System, method and computer program product for performing a contingent claim valuation of an early-launch option

ABSTRACT

A system, method and computer program product are provided for performing a contingent claim valuation of an early-launch option including a contingent claim exercisable at one of a plurality of decision points including one or more decision points before an expiration exercise point. The method may include determining first and second values representing payoffs attributable to exercise of a contingent claim at a selected decision point before the expiration exercise point, and at the expiration exercise point, respectively. The first and/or second values may be determined based upon respective present value distribution of contingent future value and a respective present value of an exercise price, the present values including a respective distribution and exercise price discounted according to first and second discount rates, respectively. A value of the contingent claim may then be determined based upon the first value and/or the second value.

RELATED APPLICATIONS

The present application is a continuation-in-part of U.S. patentapplication Ser. No. 10/309,659, entitled: Systems, Methods and ComputerProgram Products for Performing a Contingent Claim Valuation, filed Dec.4, 2002, now U.S. Pat. No. 7,676,416 which is a continuation-in-part ofapplication Ser. No. 09/902,021 filed Jul. 10, 2001, now U.S. Pat. No.6,862,579, entitled: Systems, Methods and Computer Program Products forPerforming a Generalized Contingent Claim Valuation, issued Mar. 1,2005; and a continuation-in-part of U.S. patent application Ser. No.10/453,396, entitled: Systems, Methods and Computer Program Products forModeling Uncertain Future Benefits, filed Jun. 3, 2003, the contents ofall of which are hereby incorporated by reference in their entireties.

FIELD OF THE INVENTION

Embodiments of the present invention generally relate to contingentclaim valuation and, more particularly, to a system, method and computerprogram product for evaluating contingent claims of an early-launchoption.

BACKGROUND OF THE INVENTION

It is oftentimes desirable to determine the value of a contingent claimthat may be exercised at some time in the future. The two most commonforms of a contingent claim are a call and a put, both of which mayarise in a wide variety of applications. For example, financial optionscommonly involve a call in which a stock or other financial instrumentmay be purchased at some time in the future for a predetermined exerciseprice or a put in which a stock or other financial instrument may besold at some time in the future for a predetermined exercise price.While contingent claims frequently occur in the financial arena,contingent claims also arise in a number of other contexts, such asproject evaluation and the evaluation of options to purchase or sellother assets, as described below. Unfortunately, the contingent claimsthat arise in these other contexts may be more difficult to evaluatethan the contingent claims that arise in the financial context since theunderlying assets in these other contexts are not traded or valued by awell established market, such as the stock market in the financialarena.

By way of example of the contingent claims that occur in contexts otherthan the financial arena, the contingent claims that arise duringproject evaluation and options to purchase or sell other assets will behereinafter described. In this regard, a number of projects arestructured so as to include a contingent claim that may be exercised byone of the participants at some time in the future. The contingent claimoftentimes comes in the form of a call in which one of the participantshas an option to invest additional amounts of money in order to continuethe project. As such, if the initial stages of the project have provedunsuccessful and/or if the future prospects for the project appearbleak, the participant capable of exercising the call will likelydecline to invest additional money and thereby forego exercise of thecall and will therefore terminate its participation in the project.Alternatively, if the initial stages of the project have been successfuland/or if the prospects of success of the project are bright, theparticipant capable of exercising the call will likely make thenecessary investment in order to continue its participation in theproject.

The investment that guarantees continued participation will often takethe formal financial form of a purchase of another call option.Concretely, the investment may be used to fund additional engineeringand/or market research that continues the intermediate development ofthe technology or product. At these intermediate stages of development,there may not yet be sufficient confidence to commit to full-scaleproduction, which may eventually proceed contingent on successfuldevelopmental progress, or, if unsuccessful, then termination of theeffort. Therefore, funding for development may occur as a small, phasedincremental stream of investments the values of which are calculated asa series of contingent claims. In sum, technology or product developmentcan be modeled as a succession of call options.

Examples of projects that include a contingent claim at some subsequenttime are widely varied, but one common example involves a project havinga pilot phase extending from some initial time to a subsequent time atwhich the contingent claim may be exercised. If the contingent claim isexercised, such as by one of the participants contributing the necessaryinvestment to the project, the project will enter a commercial phase. Asa more specific example, the project may involve research anddevelopment having staged investments in which each investment isessentially a contingent claim with the participant opting to continuewith the research and development activity if the participant makessufficient progress and the necessary investment, but withdrawing fromthe research and development activity if the participant declines tomake the investment.

By way of other specific examples, the contingent claim may represent anoption for the participant to adjust its production level at asubsequent time or an option to adjust its production mix in the future.In such examples, not all the complete fabrication equipment or factoryreal estate is purchased at the onset of production, but rather asmarket demand proves strong, additional equipment is installed. Theresult is lowered risk in the face of uncertain market demand, which isoffset by making a series of contingent, incremental investments. Thedifficult question is how to appropriately size and time investmentsbalancing market uncertainty, the benefit of being well-equipped or theregret of having prematurely invested.

In addition to project analysis, contingent claims may arise in thecontext of contingent clauses in contractual agreements that may takeadvantage of incremental changes in performance contingent on specifiedanticipated future events. In such instances, shifts in deliveryquantities or sales quotas can be set against a series of incrementalinvestments or payments the amount of which is calculated as amulti-stage or compound option.

And in yet other scenarios, contingent claims may arise in the contextof an option to purchase or sell assets other than financial assets. Insuch contexts, the contingent claim oftentimes comes in the form of acall or a put in which one of the participants purchases the contingentclaim to thereby have an option to purchase an asset or sell an asset atsome subsequent time for a predetermined exercise price. The asset insuch contexts can comprise any of a number of different assets, bothtangible and intangible assets, including goods, services, and licensessuch as cruise ship tickets, tickets to the theatre or a sporting event,the rental of a hotel room, and the rental of a car. In a more specificexample, then, the contingent claim may comprise an option to purchasean airline ticket with the option being purchased at some initial time,and the option capable of being exercised at a subsequent time topurchase the airline ticket.

In another similar example, the contingent claim may comprise an optionto obtain a full refund on an asset purchased at some initial time, withthe option being exercisable at a subsequent time to obtain a fullrefund. In a more specific example, the asset may comprise an airlineticket purchased at some initial time, where the airline ticket ispurchased with an option to obtain a refund of the purchase price at asubsequent time at which the option may be exercised. If the option, orcontingent claim, is exercised, the purchaser will then be able toobtain a refund of the purchase price of the ticket by selling theticket back to the airline ticket vendor (e.g., airline).

Regardless of the type of contingent claim, it is desirable to determinethe value of a project and, in particular, the contingent claim at thepresent time. By determining the value of the contingent claim, theparticipant can avoid overpaying for the project or asset as a result ofan overvaluation of the contingent claim. Conversely, the participantcan identify projects or assets in which the value of the contingentclaim has been undervalued and can give strong consideration toinvesting in these projects or assets since they likely representworthwhile investment opportunities. And although techniques have beendeveloped for determining the value of a project or an asset having acontingent claim at one or more subsequent times, it is usuallydesirable to improve upon existing techniques.

SUMMARY OF THE INVENTION

In view of the foregoing background, exemplary embodiments of thepresent invention provide an improved system, method and computerprogram project for performing a contingent claim valuation of anearly-launch option. According to one aspect of exemplary embodiments ofthe present invention, a method is provided for performing a contingentclaim valuation of an early-launch option including a contingent claimexercisable at one of a plurality of decision points (e.g., p_(n), n=1,2, . . . N≦T). The decision points include an expiration exercise point(e.g., p_(N) (t=T)) and one or more decision points (e.g., p_(n), n<N)before the expiration exercise point.

The method of one exemplary embodiment includes determining a firstvalue (e.g., S_(T) Payoff=S_(p) ₁ e^(−r) ¹ ^(p) ¹ −x_(T)e^(−r) ² ^(p) ¹) representing a payoff attributable to exercise of a contingent claimat a selected decision point before the expiration exercise point, anddetermining a second value (e.g., S_(T) Payoff=E[max(S_(T)e^(−r) ¹^(T)−x_(T)e^(−r) ² ^(T),0)]) representing a payoff attributable to theexercise of a contingent claim at the expiration exercise point. In thisregard, the first and/or second values may be determined based uponrespective present value distribution of contingent future value and arespective present value of an exercise price. The present valuedistribution may include a respective distribution of contingent futurevalue discounted according to a first discount rate (e.g., r₁), and thepresent value of the exercise price may include a respective exerciseprice discounted according to a second discount rate (e.g., r₂) thatneed not equal the first discount rate. More particularly, for example,the first value may be determined based upon a present valuedistribution of contingent future value at the selected decision point(e.g., S_(p) ₁ e^(−r) ¹ ^(p) ¹ ), and based upon a present value of anexercise price at the selected decision point (e.g., x_(T)e^(−r) ² ^(p)¹ ).

Additionally or alternatively, the second value may be determined basedupon a present value distribution of contingent future value at theexpiration exercise point (e.g., S_(T)e^(−r) ¹ ^(T)), and based upon apresent value of an exercise price at the expiration exercise point(e.g., x_(T)e^(−r) ² ^(T)). Even more particularly, for example,determining the second value may include determining an expected valueof the difference between the present value distribution of contingentfuture value at the expiration exercise point and the present value ofthe exercise price at the expiration exercise point (e.g., E[S_(T)e^(−r)¹ ^(T)−x_(T)e^(−r) ² ^(T)]). And further, determining the expected valueof the difference may include limiting the difference to a minimumpredefined value (e.g., E[max(S_(T)e^(−r) ¹ ^(T)−x_(T)e^(−r) ²^(T),0)]).

The method of the aforementioned aspect also includes determining avalue of the contingent claim (e.g., S_(T) Payoff) based upon the firstvalue and/or the second value. The value of the contingent claim may bedetermined further based upon a milestone threshold at the selecteddecision point (e.g., P*p₁), where the milestone threshold represents aminimum asset value for exercising the contingent claim at the selecteddecision point. More particularly, for example, determining the valuemay include conditioning the value on a comparison of a distribution ofcontingent future value at the selected decision point and the milestonethreshold (e.g., S_(p) ₁ ≧P*p₁).

The milestone threshold (e.g., P*p₁) for a selected decision point maybe determined in a number of different manners in accordance with anumber of different techniques. In one technique, for example,determining the milestone threshold may include determining a presentvalue conditional distribution of contingent future value attributableto the exercise of a contingent claim at the expiration exercise point(e.g., S_(T|P*p) ₁ e^(−r) ¹ ^(T)), where the present value conditionaldistribution includes a distribution of contingent future valueconditioned on an estimated milestone threshold, and discountedaccording to the first discount rate. In addition, this technique mayinclude determining present values of respective exercise pricesrequired to exercise one or more contingent claims at the expirationexercise point and the selected decision point (e.g., x_(T)e^(−r) ² ^(T)and x_(T)e^(−r) ² ^(p) ¹ ), where the present values of respectiveexercise prices include respective exercise prices discounted accordingto the second discount rate. A conditional value (e.g., Net S_(T|P*p)_(n) Payoff) may then be determined based upon the present valueconditional distribution of contingent future value and the presentvalues of respective exercise prices. To realize a milestone threshold,a present value conditional distribution and a conditional value may berepeatedly determined for different estimated milestone thresholds untilthe conditional value equals approximately zero (e.g., Net S_(T|P*p)_(n) Payoff≈0). The respective milestone threshold at that instance maythen be selected as the milestone threshold for the selected decisionpoint.

According to another technique, determining the milestone threshold mayinclude determining a first value and a second value, and determining avalue based upon the first value and/or the second value, where thisvalue is conditioned on a comparison of a distribution of contingentfuture value at the selected decision point and a candidate milestonethreshold (e.g., S_(p) ₁ ≧P*p_(1,j)). In this regard, a value may berepeatedly determined for a plurality of candidate milestone thresholds(e.g., P*p_(1,j), j=1, 2, . . . J) such that a candidate milestonethreshold that maximizes the value may be selected as the milestonethreshold for the selected decision point.

In accordance with yet another technique, determining the milestonethreshold may include determining a present value of an underlying assetat a selected decision point before the expiration exercise point (e.g.,P*p_(1,j)e^(−r) ¹ ^(T)), where the present value of the underlying assetincludes a candidate milestone threshold discounted according to thefirst discount rate. This technique may then include determining a thirdvalue (e.g., P*p_(1,j) Payoff=P*p_(1,j)e^(−r) ¹ ^(p) ¹ −x_(T)e^(−r) ²^(p) ¹ ) based upon the present value of the underlying asset, and basedupon a present value of an exercise price at the selected decision point(e.g., x_(T)e^(−r) ² ^(p) ¹ ). A present value of an underlying assetand a first value may be determined for one or more candidate milestonethresholds for the selected decision point (e.g., P*p_(1,j), j=1, 2, . .. J). Accordingly, a candidate milestone threshold may be selected for athird value approximately equal to a second value, where the respectivemilestone threshold is selected as the milestone threshold for theselected decision point.

According to other aspects of exemplary embodiments of the presentinvention, an improved system and computer program product are providedfor performing a contingent claim valuation of an early-launch option.Exemplary embodiments of the present invention therefore provide animproved system, method and computer program product for performing acontingent claim valuation of an early-launch option. As indicated aboveand explained in greater detail below, the system, method and computerprogram product of exemplary embodiments of the present invention maysolve the problems identified by prior techniques and may provideadditional advantages.

BRIEF DESCRIPTION OF THE DRAWINGS

Having thus described the invention in general terms, reference will nowbe made to the accompanying drawings, which are not necessarily drawn toscale, and wherein:

FIG. 1 is a flowchart including various steps in a method of performinga contingent claim valuation of a multi-stage option, according to oneexemplary embodiment of the present invention;

FIGS. 2 a-2 d are flowcharts including various steps in a method ofdetermining a milestone threshold for performing a multi-stage optionvaluation in accordance with a “benefit-regret” technique, “arc”technique, “zero crossing” technique and “sorted list” technique,respectively, according to exemplary embodiments of the presentinvention;

FIG. 3 illustrates exemplary distributions of contingent future valuedefined for a number of exercise points for performing a multi-stageoption valuation, in accordance with exemplary embodiments of thepresent invention;

FIG. 4 illustrates an exemplary distribution of contingent futurebenefits along with an estimated milestone threshold and conditionaldistribution of contingent future benefits for performing a multi-stageoption valuation, in accordance with exemplary embodiments of thepresent invention;

FIG. 5 illustrates, for the distributions of FIG. 4, two of a number ofdifferent conditional paths the future value may take from one exercisepoint to another for performing a multi-stage option valuation, inaccordance with exemplary embodiments of the present invention;

FIG. 6 illustrates a distribution of net conditional payoff values for anumber of calculated net conditional payoff values for performing amulti-stage option valuation (conditioned on a milestone threshold at anext-to-last exercise point), in accordance with exemplary embodimentsof the present invention;

FIG. 7 illustrates two of a number of different the conditional pathsthat conditional future value may take from one exercise point toanother for three different candidate milestone thresholds forperforming a multi-stage option valuation, in accordance with exemplaryembodiments of the present invention;

FIG. 8 illustrates a number of different conditional paths thatconditional future value may take from one exercise point to another,and if appropriate through that exercise point, for an estimatedthreshold for performing a multi-stage option valuation, in accordancewith exemplary embodiments of the present invention;

FIG. 9 illustrates a distribution of net conditional payoff values for anumber of calculated net conditional payoff values for performing amulti-stage option valuation (conditioned on a milestone threshold at apreceding exercise point), in accordance with exemplary embodiments ofthe present invention;

FIGS. 10 a and 10 b are graphs plotting a number of payoff values for anumber of candidate milestone thresholds for respective exercise points,and including a selected milestone threshold associated with a maximummean net payoff value for performing a multi-stage option valuation, inaccordance with exemplary embodiments of the present invention;

FIGS. 11 a and 11 b are scatter plots of a number of mean net payoffvalues for a number of candidate milestone thresholds for respectiveexercise points, and including an exemplary quadratic function definedbased thereon, for performing a multi-stage option valuation, inaccordance with exemplary embodiments of the present invention;

FIG. 12 is a graph plotting a number of payoff values for a number offorecasted asset values for respective exercise points, and including aselected milestone threshold associated with a maximum mean net payoffvalue for performing a multi-stage option valuation, in accordance withexemplary embodiments of the present invention;

FIG. 13 illustrates a number of different conditional paths thatconditional future value may take from one exercise point to another,and if appropriate through that exercise point, for an initial meanvalue of an asset, for performing a multi-stage option valuation, inaccordance with exemplary embodiments of the present invention;

FIG. 14 illustrates a distribution of payoff values for a number ofcalculated payoff values for valuation of a multi-stage option, inaccordance with exemplary embodiments of the present invention;

FIG. 15 is a flowchart including various steps in a method of performinga contingent claim valuation of an early-launch option, according toexemplary embodiments of the present invention;

FIGS. 16 a-16 d are flowcharts including various steps in a method ofdetermining a milestone threshold for performing an early-launch optionvaluation in accordance with a “benefit-regret” technique, “arc”technique, “sorted list” technique and “conditional” technique,respectively, according to exemplary embodiments of the presentinvention;

FIG. 17 illustrates exemplary distributions of contingent futurebenefits defined for a payout point and expiration exercise point forperforming an early-launch option valuation, in accordance withexemplary embodiments of the present invention;

FIG. 18 illustrates an exemplary distribution of contingent futurebenefits along with an estimated milestone threshold and conditionaldistribution of contingent future benefits for performing anearly-launch option valuation, in accordance with exemplary embodimentsof the present invention;

FIG. 19 illustrates, for the distributions of FIG. 18, two of a numberof different conditional paths the future value may take from the payoutpoint to the expiration exercise point for performing an early-launchoption valuation, in accordance with exemplary embodiments of thepresent invention;

FIG. 20 illustrates a distribution of net conditional payoff values fora number of calculated net conditional payoff values for performing anearly-launch option valuation (conditioned on a milestone threshold atthe payout point), in accordance with exemplary embodiments of thepresent invention;

FIG. 21 illustrates two of a number of different the conditional pathsthat conditional future value may take from the payout point to theexpiration exercise point for three different candidate milestonethresholds for performing an early-launch option valuation, inaccordance with exemplary embodiments of the present invention;

FIG. 22 is a graph plotting a number of payoff values for a number ofcandidate milestone thresholds for the payout point, and including aselected milestone threshold associated with a maximum mean net payoffvalue for performing an early-launch option valuation, in accordancewith exemplary embodiments of the present invention;

FIG. 23 is a graph plotting conditional payoffs at the expiration pointand early exercise payoffs at the payout point for a number of candidatemilestone thresholds, and identifying a milestone threshold for whichthe conditional payoff and early exercise payoff are approximatelyequal, for performing an early-launch option valuation, in accordancewith exemplary embodiments of the present invention;

FIG. 24 illustrates a number of different conditional paths thatconditional future value may take from an initial time to the payoutpoint, and if appropriate through the payout point to the expirationpoint, for an initial mean value of an asset, in accordance withexemplary embodiments of the present invention;

FIG. 25 illustrates a distribution of payoff values for a number ofcalculated payoff values for valuation of an early-launch option, inaccordance with exemplary embodiments of the present invention;

FIG. 26 is a flowchart including various steps in a method of performinga contingent claim valuation of a combination option, according toexemplary embodiments of the present invention; and

FIG. 27 is a schematic block diagram of the system of one exemplaryembodiment of the present invention embodied by a computer.

DETAILED DESCRIPTION OF THE INVENTION

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which preferred embodimentsof the invention are shown. This invention may, however, be embodied inmany different forms and should not be construed as limited to theembodiments set forth herein; rather, these embodiments are provided sothat this disclosure will be thorough and complete, and will fullyconvey the scope of the invention to those skilled in the art.

Exemplary embodiments of the present invention provide a system, methodand computer program product for performing a valuation of a contingentclaim, such as a call option or a put option, at a time prior toexercise of the contingent claim. The system, method and computerprogram product of exemplary embodiments of the present invention willbe described in conjunction with the valuation of a contingent claim atan initial, present time. However, in determining the present value of acontingent claim, the system, method and computer program product ofexemplary embodiments of the present invention may be capable ofdetermining the present value of the contingent claim at any time priorto the exercise of the contingent claim such that subsequent discussionof present value therefore including a valuation at any time prior tothe exercise of the contingent claim.

The system, method and computer program product may be utilized toperform a valuation of a variety of contingent claims. These contingentclaims may be either calls or puts, although calls will be discussedhereinafter by way of example. In addition, the contingent claims mayarise in a variety of contexts. For example, the contingent claim mayinvolve the exercise of a real option, that is, an option that may beexercised at one or more points in time in the future in order toexploit or to continue to exploit an asset or activity, as opposed to afinancial asset. In this regard, the real option may arise during aproject analysis as discussed in detail below for purposes of example.However, the contingent claim may involve the exercise of other types ofoptions, including financial options. In this regard, the system, methodand computer program product of the present invention may provideadvantages relative to the Black-Scholes method even in the context ofevaluating financial options since the methodology of the presentinvention is not constrained by the assumptions upon which theBlack-Scholes formula is premised.

Even with respect to project analysis, however, the system, method andcomputer program product are capable of performing a valuation of thecontingent claims present in a wide variety of projects. In this regard,the project may have a pilot phase extending from some initial time toone or more subsequent points in time at which one or more contingentclaims are to be exercised. If the contingent claim is exercised, suchas by one of the participants contributing the necessary investment tothe project, the project of this example will continue and mayultimately enter a commercial phase. As a more specific example, theproject may involve research and development having staged investmentsin which each investment is essentially a contingent claim with theparticipant opting to continue with the research and developmentactivity if the participant makes the necessary investment, butwithdrawing from the research and development activity if theparticipant declines to make the investment. By way of other specificexamples, the contingent claim may represent an option for theparticipant to adjust its production level at a subsequent time or anoption to adjust its production mix in the future.

In addition to project analysis, the contingent claim may arise in thecontext of evaluation of an option to purchase or sell an asset, eitherin or out of the financial arena. In such contexts, the system, methodand computer program product are capable of performing a valuation ofthe contingent claims for the purchase of a wide variety of assets. Inthis regard, the contingent claim may comprise an option to purchase anasset at a subsequent time at which the option is to be exercised, wherethe contingent claim is purchased at an initial time prior to theexercising the option. In a more specific example, then, the contingentclaim may comprise an option to purchase an airline ticket with theoption being purchased at some initial time, and the option capable ofbeing exercised at some subsequent time to purchase of the airlineticket.

In another similar example, the contingent claim may comprise an optionto obtain a full refund on an asset purchased at some initial time, withthe option being exercisable at a subsequent time to obtain a fullrefund. In a more specific example, the asset may comprise an airlineticket purchased at some initial time, where the airline ticket ispurchased with the option to obtain a refund of the purchase price at asubsequent time at which the option may be exercised. If such an option,or contingent claim, is exercised, the purchaser will then be able toobtain a refund of the purchase price of the ticket by selling theticket back to the airline ticket vendor (e.g., the airline).

By way of other specific examples, the contingent claim may represent anoption for the participant to purchase any of a number of differentassets, particularly in instances in which the value of the asset (i.e.,price consumers are willing to pay for the asset) can vary over time orbetween purchases, such as in the case of the purchase of cruise shiptickets, the purchase of tickets to the theatre or a sporting event, therental of a hotel room, and the rental of a car. While various exampleshave been provided, it should be understood that the system, method andcomputer program product may be utilized to evaluate a number of othercontingent claims, in the project analysis context, in the context ofevaluation of an option to purchase or sell an asset (both in and out ofthe financial arena), and in other contexts, if so desired.

In accordance with exemplary embodiments of the present invention, thecontingent-claim valuation may be performed based upon the futurebenefits that may flow from an underlying asset. The future benefits mayinclude, for example, the gross operating profit associated with anasset or project following the exercise of a contingent claim at somesubsequent time, t. Alternatively, for example, the future benefits mayinclude the value associated with an asset following the exercise of acontingent claim at some subsequent time, t, where the value isdetermined based upon a price consumers are willing to pay for the assetless the cost of materials and labor. In the example in which thecontingent claim is an option to purchase a ticket, the future benefitsmay include the ticket prices that consumers would be willing to pay atthe time of exercising the option. It should be understood, however,that the future benefits may represent a wide variety of other types offuture benefits depending upon the context.

In addition to future benefits, the contingent-claim valuation ofexemplary embodiments of the present invention may be performed basedupon the price of exercising the option at one or more points over theperiod of time. In the context of a call option, for example, anexercise price may be referred to as a contingent future investment andinclude the cost or purchase price of the contingent claim or call atsome subsequent time t.

In accordance with exemplary embodiments of the present invention, thereal option may comprise a “go, no-go” (or “traversal” or European)option including a contingent claim at a single, predefined exercisepoint, or an “early-launch” (or American) option including a contingentclaim at a single, variable exercise point. Also, for example, the realoption may comprise a “multi-stage” (or compound) option including aplurality of contingent claims at a plurality of exercise points over aperiod of time, where the exercise points may be predefined (go, no-go)and/or variable (early-launch). And in yet a further example, the realoption may comprise a combination option comprising some combination ofa go, no-go option, early-launch option and/or compound option.Performing a contingent-claim valuation in the context of a compoundoption, early-launch option and combination option, in accordance withexemplary embodiments of the present invention, is explained in greaterdetail below. For further information on performing a contingent-claimvaluation in the context of a go, no-go option, in accordance withexemplary embodiments of the present invention, see the aforementionedU.S. patent application Ser. No. 10/309,659, and U.S. Pat. No.6,862,579.

A. Multi-Stage Option

In accordance with one exemplary embodiment of the present invention, acontingent-claim valuation may be performed for a compound ormulti-stage option including a plurality of exercise points over aperiod of time. Valuation of the multi-stage option will be describedherein with reference to predefined (go, no-go) exercise points. Itshould be understood, however, that valuation of the multi-stage optionmay be equally performed with reference to variable (early-launch)exercise points.

Referring to FIG. 1, a method of performing a contingent claim valuationof a multi-stage option according to one exemplary embodiment of thepresent invention may begin by defining a period of time and theexercise points within that period of time, as shown in block 10. Inthis regard, the period of time can begin at t=0 and extend to t=T. Theperiod of time can then be divided into a number of different timesegments. Within the period of time, at least some of the time segmentsmay correspond to respective exercise points, where an exercise point att=T may be referred to as a final, or expiration, exercise point, andwhere the exercise points may be more generally referred to as “decisionpoints” within the period of time. In one embodiment, for example, thetime period T is defined such that each time segment and exercise pointcan be represented as an integer divisor of T, i.e., t=0, 1, 2, . . . T;and the number N of exercise points may be defined as p_(n), n=1, 2, . .. N≦T, where each p_(n) corresponds to a time segment of the period oftime. Thus, for example, the period of time can be defined as a numberof years (e.g., T=5) divided into a number of one-year time segmentswhich, including the initial time t=0, totals the number of years plusone time segment (e.g., t=0, 1, 2, . . . 5). As used herein, each timesegment begins at time t and ends at time t+1 (presuming the timesegment is an integer divisor of T), and is defined by the beginningtime t. Thus, time segment t=1 extends from time t=1 to time t=2.Similarly, time segment t=2 extends from t=2 to t=3. For an example ofthe time segments for a period of time, as well as the exercise pointswithin that period of time, see Table 1 below.

TABLE 1 First Discount Rate 12% Second Discount Rate 5% Time Segment 0 12 3 4 5 Exercise Point p₁ = 1 p₂ = 3 Expiration (p₃ = 5) Uncertainty 40%40% 40% 40% 40% 40% Exercise Price $15.00 $30.00 $120.00

Before, during or after defining the time period, a number of parametersmay be selected, determined or otherwise calculated for subsequent usein performing a contingent-claim valuation in accordance with thisexemplary embodiment of the present invention, as shown in block 12. Anumber of these parameters, including first and second discount ratesfor determining or otherwise discounting future benefits and exerciseprices at one or more exercise points, and uncertainties and exerciseprices at the exercise point time segments, are described below, withexamples provided in Table 1.

First and second discount rates, r₁ and r₂, which at different instancesmay be referred to or otherwise function as interest or growth rates,may be selected in any of a number of different manners. The firstdiscount rate may be selected to take into account the risk associatedwith future benefits. In some embodiments, for example, the firstdiscount rate comprises the weighted average cost of capital (WACC)since the WACC may provide an average discount rate with which manyanalysts are familiar. The second discount rate, on the other hand, maybe selected to take into account the risk associated with the contingentclaim or exercise price as known to those skilled in the art. In someembodiments, for example, the second discount rate comprises the riskfree rate of discounting. In other embodiments, however, the exerciseprice may be subject to a non-market or corporate risk such that theappropriate second discount rate may be the corporate bond rate.

The first and second discount rates may be equal or different from oneanother. In this regard, selecting equal first and second discount ratesmay, for example, define a risk-neutral condition. Alternatively, forexample, selecting a second discount rate less than the first discountrate may define a risk-averse condition. And although not describedherein, a second discount rate may be selected to be greater than thefirst discount rate, if so desired. Further, for example, the first andsecond discount rates may be selected or otherwise set for the entireperiod of time, for each segment of the period of time, or moreparticularly for each exercise point within the period of time. Thus,respective first and/or second discount rates may remain the same orchange from one time segment to the next, or more particularly from oneexercise point to the next.

In addition to selecting the discount rates, uncertainty, or volatility,in the market including the asset may be selected or otherwisedetermined for each exercise point p_(n), or more generally for eachsegment of the period of time t. The uncertainty may be selected ordetermined in any of a number of different manners, and may be the sameor different from one segment to the next. In one exemplary embodiment,for example, the uncertainty may be determined based upon a model ofreturns, or growth rate (e.g., first discount rate), versus risk, oruncertainty, such as in the manner disclosed in the aforementioned U.S.patent application Ser. No. 10/453,396. More particularly, in oneembodiment for example, the returns may be modeled from two risk valuesand associated return values, as such may be determined by an estimatoror the like. Then, assuming a typical linear relationship between riskand return (sometimes referred to as the CAPM or Capital Asset PricingModel), the risk can be modeled as a linear function of returns basedupon the two risk values and associated return values. For example,according to one exemplary embodiment, two uncertainty risk values maycomprise 20% and 30%, with associated return values comprising 10.0% and12.5%, respectively. With such values, risk can be modeled as a linearfunction of return as follows:Risk(Return)=4×(Return−5)where return and risk are expressed as percentages. For a furtherdescription of modeling risk as a function of returns, see U.S. patentapplication Ser. No. 10/453,395, entitled: Systems, Methods and ComputerProgram Products for Modeling a Monetary Measure for a Good Based UponTechnology Maturity Levels, filed Jun. 3, 2003, the content of which isincorporated by reference in its entirety.

Further to selecting the discount rates and selecting or otherwisedetermining the uncertainty, exercise prices x_(p) _(n) may be selectedor otherwise determined for the exercise points p_(n). These exerciseprices may be selected in any of a number of different manners, such asin accordance with any of a number of different conventional projectanalysis techniques. One or more of the exercise prices may eachcomprise a single payment that has a predetermined value. In variousinstances, however, one or more of the exercise prices may not have asingle value, but may be best represented by a distribution of exerciseprices that relate probabilities to each of a plurality of differentexercise prices. For more information on such instances, see theaforementioned U.S. patent application Ser. No. 10/309,659, and U.S.Pat. No. 6,862,579.

Also before, during or after defining the time period, an initial, meanvalue of the asset may, but need not, be defined for the initial timesegment (t=0), as shown in block 14. If so desired, the initial assetvalue can be defined in any of a number of different manners. Forexample, in one embodiment, the initial value can be defined as thepresent value of a forecasted market. For example, in another relatedembodiment, the initial value can be defined as the current value of apotential future product or technology. For a description of onetechnique of determining the maximum gross profitability, see U.S.patent application Ser. No. 10/453,727, entitled: Systems, Methods andComputer Program Products for Modeling Demand, Supply and AssociatedProfitability of a Good, filed Jun. 3, 2003, the content of which ishereby incorporated by reference in its entirety.

In addition, before, during or after defining the time period, a revenueor value distribution S can be determined or otherwise calculated foreach exercise point p_(n), as shown in block 16. In this regard, eachvalue distribution may be considered a distribution of contingent futurevalue (asset value) attributable to exercising the contingent claim at arespective exercise point. And more particularly, the value distributionat the expiration exercise point p_(N) (t=T) may be considered adistribution of contingent future benefits attributable to exercisingthe contingent claim at the expiration exercise point. Each distributionof contingent future value S_(p) _(n) may be determined in any of anumber of different manners but, in one exemplary embodiment, isdetermined based upon the mean asset value at the respective exercisepoint and the standard deviation in time at the exercise point. Thedistributions S_(p) _(n) , and more particularly their mean values andstandard deviations, may be derived in accordance with a continuous-timetechnique, or alternatively in accordance with a discrete-timetechnique. For one exemplary continuous-time technique for determiningthe mean value and standard deviation for defining a distribution S_(p)_(n) , see the aforementioned U.S. patent application Ser. No.10/453,396. More particularly, and in accordance with another technique,the mean asset value at each exercise point may be determined asfollows:μ_(S) _(pn) =μ₀ ×e ^(r) ¹ ^(p) ^(n)   (1)In equation (1), μ_(S) _(pn) represents the mean asset value at thecurrent exercise point p_(n), μ₀ represents the initial mean asset valueat the initial time segment (t=0), and r₁ represents the first discountrate (growth rate in this instance). Continuing the example of Table 1above, see Table 2 for an example of the initial mean value and the meanvalues at each of the exercise point p_(n), n=1, 2, 3 (t=1, 3, 5).

TABLE 2 Initial Mean Value $100.00 Time Segment 0 1 2 3 4 5 ExercisePoint p₁ = 1 p₂ = 3 Expiration (p₃ = 5) Mean Value $100.00 $112.75$143.33 $182.21 Standard Deviation $0.00 $46.97 $112.50 $201.72Correlation Coefficient 0.58 0.77 0.45 (p₁, p₃) (p₅, p₃) (p₁, p₅)

In accordance with one technique, the standard deviation may bedetermined as follows:σ_(S) _(pn) =μ_(S) _(pn) ×√{square root over (e ^(u) ^(n) ² ^(×p) ^(n)−1)}  (2)In equation (2), σ_(S) _(pn) represents the standard deviation forexercise point p_(n); and u_(n) represents the uncertainty value at therespective exercise point, where the uncertainty may be represented as adecimal. For an example of the standard deviations for exercise pointsp_(n), n=1, 2, 3 (t=0, 1, 3, 5), see Table 2.

As indicated above, the distributions S_(p) _(n) may be alternativelydetermined in accordance with a discrete-time technique. For oneexemplary discrete-time technique for determining the mean value andstandard deviation for defining a distribution S_(p) _(n) , see theaforementioned U.S. patent application Ser. No. 10/453,395. Inaccordance with such techniques, an initial, mean asset value need notbe defined for the initial time segment (t=0). Instead, the mean andstandard deviation for defining a distribution at an exercise point maybe determined based upon a quantitative measure of maturity of atechnology of the asset (the measure of maturity being associated with arisk/return distribution), as well as a mean value associated with thetechnology.

After determining the mean asset value and standard deviation, a valuedistribution of contingent future value S_(p) _(n) can be determined forthe respective exercise point p_(n) by defining each distribution ofcontingent future value according to the respective mean asset value andstandard deviation. The distribution of contingent future value can berepresented as any of a number of different types of distributions but,in one embodiment, the distribution of contingent future value isdefined as a lognormal distribution. In this regard, FIG. 3 illustratesdistributions of contingent future value S_(p) _(n) defined for exercisepoints p_(n), n=1, 2, 3 (t=1, 3, 5) for the example in Tables 1 and 2.

Further to determining distributions of contingent future value,relationships between respective distributions, and in particularbetween respective discrete-time distributions, for example, may beestablished. These relationships may be established in a number ofdifferent manners, such as via a correlation coefficient, as is known tothose skilled in the art. The correlation coefficient may be selected ordetermined in any one of a number of manners. In one embodiment, forexample, the correlation coefficient (Coeff_(p) _(a) _(,p) _(b) )between a distribution of contingent future value at a particularexercise point p_(a) (i.e., S_(p) _(a) ), and the distribution ofcontingent future value at a another exercise point p_(b) (i.e., S_(p)_(b) ), may be determined as follows:

$\begin{matrix}{{Coeff}_{p_{a},p_{b}} = \sqrt{\frac{u_{a}^{2}p_{a}}{u_{b}^{2}p_{b}}}} & (3)\end{matrix}$A correlation coefficient may similarly be selected for the expirationexercise point p_(N) (t=T) being the particular exercise point p_(a),but instead of being referenced to the expiration exercise point and asubsequent exercise point in equation (3) above, the respectivecorrelation coefficient may refer back to the first exercise point oranother preceding exercise point as the particular exercise point (e.g.,p_(a)=p₁) and to the expiration exercise point as the other, subsequentexercise point (i.e., p_(b)=p_(N)). For an example of the correlationcoefficients between distributions of contingent future value at anumber of exercise points, see Table 2 above.

Irrespective of exactly how the distributions of contingent future valueS_(p) _(n) are determined, the value of the multi-stage option may bedetermined or otherwise calculated based thereon. Before determining thevalue of the multi-stage option, however, exemplary embodiments of thepresent invention may account for situations in which a reasonablyprudent participant may not exercise an option at a particular exercisepoint (and thus any remaining exercise points). More particularly,exemplary embodiments of the present invention may calculate orotherwise determine a milestone threshold for at least some, if not all,of the exercise points, where each milestone threshold represents theminimum asset value at which a reasonably prudent participant willexercise the contingent claim at that exercise point.

As will be appreciated, one of the more important tasks in data miningis to determine a milestone that distinguishes a good decision from abad decision. In the context of a contingent claim, for example, a gooddecision may be considered one whereby a decision to exercise the claimresults in a discounted asset value (benefits) greater than or equal tothe price required to achieve that value (current and any subsequentexercise prices) resulting in a minimum required rate of return on aninitial investment; and a bad decision may be considered one thatresults in an asset value less than the price required to achieve thatvalue resulting in a negative rate of return on an initial investment.To facilitate making such a decision (exercising an option), informationfrom which the decision is made, such as the asset value at an exerciseprice, may be classified as to the outcome suggested or otherwisepredicted by that information. Such prediction classifiers may includetrue positive (TP) and true negative (TN) whereby information from whichthe decision is made accurately suggests the outcome, whether a positiveoutcome (value greater than or equal to exercise price) or a negativeoutcome (value less than exercise price). In the case of a contingentclaim, for example, a true positive indicator typically leads to adecision to exercise the claim and a resulting benefit; and a truenegative indicator typically leads to a decision to not exercise theclaim and an averted failure. However, prediction classifiers may alsoinclude false positive (FP) and false negative (FN) whereby suchinformation falsely reflects the outcome. That is, information may beclassified as a false positive when that information suggests a positiveoutcome, but a negative outcome actually follows from making therespective decision, thereby resulting in a regretful decision.Similarly, information may be classified as a false negative when thatinformation suggests a negative outcome, but a positive outcome wouldactually follow from making the respective decision. But since manywould not follow a negative indicator, false negatives typically resultin decision omissions.

In view of the foregoing, performing a contingent claim valuation mayinclude determining or otherwise calculating milestone thresholds P* atthe exercise points before the expiration exercise point p_(n), n=1, 2,. . . N−1, such as to facilitate maximizing benefits (TP) and minimizeregrets (FP) and omissions FN on a risk-adjusted basis, as shown inblock 18. In this regard, the milestone threshold for an exercise pointP*p_(n) may correspond to the asset value at the respective exercisepoint likely to result in a risk-adjusted, discounted final value at theexpiration time segment substantially equal to the risk-adjusted,discounted exercise price at the current exercise point and anysubsequent exercise points. In project management, for example, it maybe desirable to have indicators or milestones that may help determinewhether or not a project is proceeding along a trajectory that is atleast equal to or exceeds a minimum rate of return for a giveninvestment. In many current project management practices, however, theestablishment of project milestones is ad hoc, often without regard torequired rates of return, particularly with respect to intermediatedecision points. It may therefore be desirable to establish a quantifiedmilestone that may indicate a minimum risk-adjusted required rate ofreturn at an intermediate decision point.

The milestone threshold for each exercise point may be determined orotherwise calculated in any of a number of different manners. Also, themilestone threshold for each exercise point may be determined in anyorder relative to the milestone threshold for any other exercise point.In one exemplary embodiment, however, the milestone thresholds may bedetermined in reverse sequential order beginning with the last exercisepoint p_(N−1) before the expiration exercise point p_(N). Further, themilestone thresholds P*p_(n) may be determined in accordance with anumber of different techniques, such as in accordance with a“benefit-regret” technique, “arc” technique, “zero crossing” techniqueor “sorted list” technique for determining a milestone threshold. Eachof the aforementioned techniques will now be described below withreference to FIGS. 2 a-2 d.

1. Benefit-Regret Technique for Determining Milestone Thresholds

More particularly, determining each milestone threshold according to thebenefit-regret technique of one exemplary embodiment of the presentinvention may include estimating a milestone threshold at the lastexercise point P*p_(N−1) before the expiration exercise pointp_(N)(t=T), as shown in FIG. 2 a, block 18 a. The milestone thresholdfor this next-to-last exercise point within the period of time may beestimated in any of a number of different manners. For example, themilestone threshold at the next-to-last exercise point may be estimatedto be approximately equal to the determined mean asset value at thatexercise point μ_(S) _(PN−1) (see equation (1)). Continuing the exampleof Tables 1 and 2, see Table 3 below for a more particular example of anestimated milestone threshold at the next-to-last exercise point p₂(t=3) (the respective milestone threshold in the example beingrepresented by P*p₂=P*3).

TABLE 3 Time Segment 0 1 2 3 4 5 Exercise Point p₁ = 1 p₂ = 3 Expiration(p₃ = 5) Estimated Threshold (P*p₂) $145.00 S_(5|P*3) Mean Value $184.33S_(5|P*3) Standard Deviation $113.20

Irrespective of exactly how the milestone threshold at the next-to-lastexercise point P*p_(N−1) is estimated, a value distribution at theexpiration exercise point p_(N) (t=T) may thereafter be determined,where the value distribution is conditioned on the estimated milestonethreshold at the next-to-last exercise point, as shown in block 18 b.This value distribution may be considered a conditional distribution ofcontingent future benefits at the expiration exercise point p_(N) (t=T),conditioned on the estimated asset value (milestone threshold) at thenext-to-last exercise point P*p_(N−1). The conditional distribution ofcontingent future benefits S_(T|P*p) _(N−1) may be determined in any ofa number of different manners. In one embodiment, for example, theconditional distribution of contingent future benefits may be determinedbased upon a conditional mean asset value at the expiration exercisepoint μ_(S) _(T|P*pN−1) and a conditional standard deviation in time atthe expiration exercise point σ_(S) _(T|P*pN−1) , such as in accordancewith the following:μ_(S) _(T|P*pN−1) =P*p _(N−1) ×e ^(r) ¹ ^((T−p) ^(N−1) ⁾  (4)σ_(S) _(T|P*pN−1) =μ_(S) _(T|P*pN−1) ×√{square root over (e ^(u) ²^(×(T−p) ^(N−1) ⁾−1)}  (5)For an example of the conditional mean value and standard deviation fora conditional distribution of future benefits at the expiration exercisepoint p_(N) (t=T) of the example of Tables 1 and 2, and the estimatedmilestone threshold at the next-to-last exercise point P*p_(N−1), seeTable 3.

After determining the conditional mean and standard deviation at theexpiration exercise point, a conditional distribution of contingentfuture benefits at the expiration exercise point S_(T|P*p) _(N−1) can bedetermined by defining the conditional distribution according to therespective mean value and standard deviation. Again, the conditionaldistribution of contingent future benefits can be represented as any ofa number of different types of distributions but, in one embodiment, theconditional distribution of contingent future benefits is defined as alognormal distribution. In this regard, see FIG. 4 for a distribution ofcontingent future benefits S_(T) at the expiration exercise point p3,(t=5), along with an estimated milestone threshold P*p_(N−1) andconditional distribution of contingent future benefits S_(T|P*p) _(N−1), for the example of Tables 1, 2 and 3.

Irrespective of exactly how the conditional distribution of contingentfuture benefits at the expiration exercise point S_(T|P*p) _(N−1) isdetermined, a conditional payoff or profit may be determined orotherwise calculated based thereon, as shown in block 18 c. Theconditional payoff can be determined in any of a number of differentmanners, including in accordance with the DM algorithm, as explained inthe aforementioned U.S. patent application Ser. No. 10/309,659, and U.S.Pat. No. 6,862,579. In accordance with the DM algorithm, the conditionaldistribution of contingent future benefits may be discounted by thefirst discount rate r₁ (e.g., WACC) to present value at t=0, and theexercise price at the expiration exercise point p_(N) (t=T) may bediscounted by the second discount rate r₂ (e.g., risk-free rate) topresent value at t=0.

In accordance with the DM algorithm, the conditional payoff at theexpiration exercise point may be determined based upon the present valueconditional distribution of contingent future benefits and the presentvalue of the exercise price at the expiration exercise point (contingentfuture investment). The conditional payoff may be determined as theexpected value of the difference between the present value conditionaldistribution of contingent future benefits and the present value of theexercise price at the expiration exercise point (e.g., contingent futureinvestment) taking into account the relative probabilities associatedwith distribution of difference values. In determining the expectedvalue of the difference between the present value distribution ofcontingent future benefits and the present value of the contingentfuture investment, a limit on the minimum permissible difference(minimum predefined value) may be established to take into account thosesituations that may exist at the time at which the contingent claim isto be exercised which would cause a participant to fail to exercise thecontingent claim. In this regard, limiting the minimum permissibledifference to zero, for example, may take into account those situationsin which the exercise of the contingent claim would otherwise create aloss since a reasonably prudent participant will fail to exercise thecontingent claim in these situations. And limiting the minimum possibledifference to values other than zero, for example, may take into accountthose situations in which reserved assets may be sold at the expirationexercise point. Thus, written notationally, the conditional payoffS_(T|P*p) _(N−1) Payoff may be determined as follows:S _(T|P*p) _(N−1) Payoff=E[max(S _(T|P*p) _(N−1) e ^(−r) ¹ ^(T) −x _(T)e ^(−r) ² ^(T),0)]  (6)In equation (6), E represents an expected value (mathematicalexpectation), and x_(T) represents the exercise price (contingent futureinvestment) at the expiration exercise point p_(N) (t=T) (e.g.,$120.00). Accordingly, a conditional payoff may be determined as afunction of conditional future benefit values s_(T|P*p) _(N−1) from theconditional distribution of contingent future benefits S_(T|P*p) _(N−1)in accordance with the following:S _(T|P*p) _(N−1) Payoff=max(s _(T|P*p) _(N−1) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)  (6a)Also, in equations (6) and (6a), “0” represents a minimum predefinedvalue of zero, although it should be understood that the minimumpredefined value may be a number of different values other than zero.

As or after determining the conditional payoff at the expirationexercise point p_(N) (t=T), the net conditional payoff may be calculatedor otherwise determined by accounting for the exercise price at thenext-to-last exercise point p_(N−1) (e.g., t=3), as shown in block 18 d.Similar to the conditional payoff, the net conditional payoff can bedetermined in any of a number of different manners. In one embodiment,for example, the net conditional payoff may be determined by discountingthe exercise price at the next-to-last exercise point by the seconddiscount rate r₂ (e.g., risk-free rate), and subtracting that discountedexercise price from the conditional payoff S_(T|P*p) _(N−1) Payoff.Written notationally, for example, the net conditional payoff NetS_(T|P*p) _(N−1) Payoff may be determined as follows:NetS _(T|P*p) _(N−1) Payoff=S _(T|P*p) _(N−1) Payoff−x _(p) _(N−1) e^(−r) ² ^(p) ^(N−1)   (7)where x_(p) _(N−1) represents the exercise price (e.g., contingentfuture investment) at the next-to-last exercise point p_(N−1) (e.g.,x_(p) _(N−1) =$30.00).

From the net conditional payoff, the mean net conditional payoff may bedetermined, as shown in block 18 e. For example, the mean netconditional payoff Mean Net S_(T|P*p) _(N−1) Payoff may be determined byselecting or otherwise forecasting a number of (e.g., 10,000)conditional future benefit values s_(T|P*p) _(N−1) from the conditionaldistribution of contingent future benefits S_(T|P*p) _(N−1) ;calculating, for those forecasted conditional future benefit values,conditional payoff and net conditional payoff values such as inaccordance with equations (6a) and (7); and calculating or otherwisedetermining the mean of the calculated net conditional payoff values. Inthis regard, the aforementioned steps may be performed to effectuateequations (6) and (7), including the expected value expression ofequation (6). The conditional future benefit values s_(T|P*p) _(N−1) ,and more generally values selected or otherwise forecasted fromdistributions as described herein, can be selected or otherwiseforecasted in any of a number of different manners. For example, thesevalues can be selected or otherwise forecasted according to a method forrandomly selecting a value from a distribution, such as the Monte Carlotechnique for randomly generating values.

FIG. 5 continues the example of FIG. 4, and illustrates two of a numberof different a conditional paths the asset value may take from thenext-to-last exercise point to the expiration exercise point for anestimated threshold, where one of the paths represents a true positive(TP) leading to a resulting benefit, and the other path represents afalse positive (FP) leading to a resulting regret. And FIG. 6 furthersthe example by illustrating a distribution of net conditional payoffvalues for a number of calculated net conditional payoff values(conditioned on a milestone threshold at the next-to-last exercise pointP*p_(N−1)).

As indicated above, the milestone threshold P*p_(n) is intended toresult in a final discounted value (payoff value) at the expirationexercise point substantially equal to the discounted exercise price atthe current exercise point and any subsequent exercise points. In otherwords, the milestone threshold is intended to result in an expected netconditional payoff value of approximately zero at time t=0, as shown inFIG. 2 a, block 18 f, and in FIG. 6. Thus, after determining the meannet conditional payoff value Mean Net S_(T|P*p) _(N−1) Payoff, if themean net conditional payoff value does not equal approximately zero,another milestone threshold P*p_(N−1) may be estimated at thenext-to-last exercise point P*p_(N−1). The method may then repeatdetermining a conditional distribution of contingent future benefitsS_(T|P*p) _(N−1) , determining a conditional payoff S_(T|P*p) _(N−1)Payoff, net conditional payoff Net S_(T|P*p) _(N−1) Payoff and mean netconditional payoff Mean Net S_(T|P*p) _(N−1) Payoff, and determining ifthe mean net conditional payoff value equals approximately zero. Themethod may continue in this manner until an estimated milestonethreshold P*p_(N−1) results in a mean net conditional payoff value equalto approximately zero. This estimated milestone threshold may then beconsidered the milestone threshold at the respective exercise point.FIG. 7 illustrates two of a number of different conditional paths futurebenefits may take from the next-to-last exercise point to the expirationexercise point for three different candidate milestone thresholds, againin the context of the example provided above.

Written in other terms, it may be said that the threshold milestoneP*p_(N−1) at the next-to-last exercise point p_(N−1) solves thefollowing expression:E[Z ₀ ^(T)(r ₁ ,r ₂)|_((s) _(T|P*pN−1=P*pN−1) ₎ ]≈x _(p) _(N−1) e ^(−r)² ^(p) ^(N−1) ,Z _(t) ₁ ^(t) ² (r ₁ ,r ₂)=E[s _(t) ₂ e ^(−r) ¹ ^((t) ² ^(−t) ¹ ⁾ −x_(t) ₂ e ^(−r) ² ^((t) ² ^(−t) ¹ ⁾]⁺In the preceding expressions, Z represents the aforementioned DMalgorithm as a function of the first and second discount rates r₁ andr₂, between two successive time segments t₁ and t₂ (e.g., t₁=0, andt₂=T), and may correspond to a discounted payoff between the respectivetime segments. Also in the expression Z, the “+” superscript representsa maximization function limiting the discounted payoff to a minimumpredefined value, such as zero. See, for example, equation (6) above.Thus, the above expression may be interpreted as solving for the casewhere the discounted payoff at the expiration exercise point (t=T),conditioned on contingent future benefits at the next-to-last exercisepoint equaling the estimated milestone threshold for that exercisepoint, equals approximately the discounted exercise price at thenext-to-last exercise point.

As shown in block 18 g, if or once the estimated milestone thresholdP*p_(N−1) results in a mean net conditional payoff value equal toapproximately zero, a milestone threshold may be estimated for anyexercise points preceding the next-to-last exercise point, such as bystarting with the exercise point p_(N−2) immediately preceding thenext-to-last exercise point p_(N−1), as again shown in block 18 a. Aswith the milestone threshold at the next-to-last exercise point, themilestone threshold at the preceding exercise point P*p_(N−2) may beestimated in any of a number of different manners. Again, for example,the milestone threshold at the preceding exercise point may be estimatedto be substantially equal to the determined mean asset at that exercisepoint, as explained above. Continuing the example of Tables 1, 2 and 3,see Table 4 below for a more particular example of an estimatedmilestone threshold at the preceding exercise point p₁ (t=1) (therespective milestone threshold in the example being represented byP*p₁=P*1).

TABLE 4 Time Segment 0 1 2 3 4 5 Exercise Point p₁ = 1 p₂ = 3 Expiration(p₃ = 5) Estimated Threshold (P*p₁) $101.00 S_(5|P*1) & S_(3|P*1)$128.40 $163.22 Mean Values S_(5|P*1) & S_(3|P*1) $78.85 $154.54Standard Dev.

Distributions of contingent future value at the expiration exercisepoint p_(N) (t=T) and the next-to-last exercise point p_(N−1) may bedetermined, where both the value distribution at the expiration exercisepoint S_(T|P*p) _(N−2) , and the value distribution at the next-to-lastexercise point S_(p) _(N−1) _(|P*p) _(N−2) , are conditioned on theestimated milestone threshold at the preceding exercise point, as againshown in block 18 b. Again, the value distributions may be considereddistributions of contingent future value at the respective exercisepoints (i.e., expiration exercise point p_(N) and next-to-last exercisepoint p_(N−1)), conditioned on the estimated value of the asset(milestone threshold) at the preceding exercise point p_(N−2). And moreparticularly, the distribution at the expiration exercise point may beconsidered a distribution of contingent future benefits attributable toexercising the contingent claim at the expiration exercise point,conditioned on the aforementioned estimated asset value.

Each of the conditional distributions of contingent future value may bedetermined in any of a number of different manners. In one embodiment,for example, the conditional distributions of contingent future valuemay be determined based upon a conditional mean asset value at therespective exercise price μ_(S) _(pn|P*pN−2) and conditional standarddeviation in time for the respective exercise price σ_(S) _(pn|P*pN−2) ,such as in accordance with the following:μ_(S) _(pn|P*pN−2) =P*p _(N−2) ×e ^(r) ¹ ^((p) ^(n) ^(−p) ^(N−2) ⁾  (8)σ_(S) _(pn|P*pN−2) =μ_(S) _(pn|P*pN−2) ×√{square root over (e ^(u) ²^(×(p) ^(n) ^(−p) ^(N−2) ⁾−1)}  (9)In the preceding equations (8) and (9), p_(n)=p_(N) (t=T) for theconditional distribution of contingent future benefits (conditionaldistribution of contingent future value) at the expiration exercisepoint; and p_(n)=p_(N−1) for the conditional distribution of contingentfuture value at the next-to-last exercise point. For examples of theconditional mean values and standard deviations for conditionaldistributions of contingent future value at the expiration exercisepoint p_(N) (t=T) and the next-to-last exercise point p_(N−1) of theexample of Tables 1, 2 and 4, and the estimated milestone threshold atthe preceding exercise point P*p_(N−2), see Table 4.

After determining the conditional means and standard deviations at theexpiration exercise point and the next-to-last exercise point,conditional distributions of contingent future value at the expirationexercise point S_(T|P*p) _(N−2) and at the next-to-last exercise pointS_(p) _(N−1) _(|P*p) _(N−2) can be determined by defining theconditional distributions according to their respective conditionalmeans and standard deviations. Similar to the others, these conditionaldistributions of contingent future value can be represented as any of anumber of different types of distributions but, in one embodiment, aredefined as lognormal distributions.

Irrespective of exactly how the conditional distributions of contingentfuture value at the expiration exercise point p_(N) (t=T) andnext-to-last exercise point p_(N−1) are determined, a conditional payoffor profit may be determined or otherwise calculated, as again shown inblock 18 c. The conditional payoff can be determined in any of a numberof different manners, including in accordance with the aforementioned DMalgorithm. In accordance with the DM algorithm, the conditionaldistribution of contingent future benefits at the expiration exercisepoint p_(N) (t=T) and the exercise price at the expiration exercisepoint may be discounted by the first discount rate r₁ and the seconddiscount rate r₂, respectively, to present value at t=0, such as in amanner similar to that described above. Then, an intermediateconditional payoff may be determined based upon the present valueconditional distribution of contingent future benefits and the presentvalue of the exercise price at the expiration exercise point, such as bydetermining the expected value of the difference therebetween, andincluding limiting the minimum permissible difference to a minimumpredefined value, such as zero.

As or after determining the intermediate conditional payoff, theconditional payoff may be determined based thereon and accounting forthe exercise price at the next-to-last exercise point p_(N−1). In thisregard, similar to above, the conditional payoff may be determined bydiscounting the exercise price at the next-to-last exercise pointp_(N−1) by the second discount rate r₂ to present value at t=0, such asin a manner similar to that described above. Then, the conditionalpayoff may be determined based upon the intermediate conditional payoffand the present value of the exercise price at the next-to-last exercisepoint, such as by determining the expected value of the differencetherebetween. Similar to limiting the minimum permissible difference,determining the conditional payoff may be further conditioned on theasset value at the next-to-last exercise point p_(N−1) (S_(p) _(N−1)_(|P*p) _(N−2) ) being at least as much as (i.e., ≧) the estimatedmilestone threshold at the next-to-last exercise point P*p_(N−1). Inthis regard, further conditioning the conditional payoff at thenext-to-last exercise point takes into account those situations in whichthe asset value at the next-to-last exercise point p_(N−1) (S_(p) _(N−1)_(|P*p) _(N−2) ) is less than the estimated milestone threshold at thenext-to-last exercise point P*p_(N−1), which may suggest a negativeoutcome (TN) and lead a reasonably prudent participant to foregoexercising the contingent claim. Thus, written notationally, theconditional payoff S_(T|P*p) _(N−2) Payoff may be determined as follows:

IF S_(p) _(N−1) _(P*p) _(N−2) ≧P*p_(N−1), thenS _(T|P*p) _(N−2) Payoff=E[max(S _(T|P*p) _(N−2) e ^(−r) ¹ ^(T) −x _(T)e ^(−r) ² ^(T),0)]−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ;else,S _(T|P*p) _(N−2) Payoff=0  (10)And as a function of conditional asset values s_(p) _(N−1) _(|P*p)_(N−2) and s_(T|P*p) _(N−2) from respective conditional distributionsS_(p) _(N−1) _(|P*p) _(N−2) and S_(T|P*p) _(N−2) , the payoff S_(T|P*p)_(N−2) Payoff may be determined as follows:

IF s_(p) _(N−1) _(|P*p) _(N−2) ≧P*p_(n−1), thenS _(T|P*p) _(N−2) Payoff=max(s _(T|P*p) _(N−2) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ;else,S _(T|P*p) _(N−2) Payoff=0

As or after determining the conditional payoff, the net conditionalpayoff may be calculated or otherwise determined by accounting for theexercise price at the preceding exercise point p_(N−2) (e.g., t=1), asagain shown in block 18 d. Similar to the conditional payoff, the netconditional payoff can be determined in any of a number of differentmanners. In one embodiment, for example, the net conditional payoff maybe determined by discounting the exercise price at the precedingexercise point by the second discount rate r₂ (e.g., risk-free rate),and subtracting that discounted exercise price from the conditionalpayoff. Written notationally, for example, the net conditional payoffNet S_(T|P*p) _(N−2) Payoff may be determined as follows:NetS _(T|P*p) _(N−2) Payoff=S _(T|P*p) _(N−2) Payoff−x _(p) _(N−2) e^(−r) ² ^(p) ^(N−2)   (11)where x_(p) _(N−2) represents the exercise price (e.g., contingentfuture investment) at the preceding exercise point p_(N−2) (e.g., x_(p)_(N−2) =$15.00).

From the net conditional payoff, the mean net conditional payoff may bedetermined, as again shown in block 18 e. For example, the mean netconditional payoff Mean Net S_(T|P*p) _(N−2) Payoff may be determined byselecting or otherwise forecasting a number of conditional asset valuesat the expiration exercise point s_(T|P*p) _(N−2) and the next-to-lastexercise point s_(p) _(N−1) _(|P*p) _(N−2) from respective conditionaldistributions of contingent future value S_(T|P*p) _(N−2) and S_(p)_(N−1) _(|P*p) _(N−2) ; calculating, for those forecasted conditionalfuture asset values, conditional payoff and net conditional payoffvalues such as in accordance with equations (10a) and (11); andcalculating or otherwise determining the mean of the calculated netconditional payoff values. Similar to the above with respect toequations (6) and (7), the aforementioned steps may be performed toeffectuate equations (10) and (11), including the expected valueexpression of equation (10).

FIG. 8 furthers the example of FIGS. 3, 4 and 5, and illustrates anumber of different a conditional paths the asset value may take fromthe preceding exercise point (p₁ in the example), through thenext-to-last exercise point (p₂ in the example) to the expirationexercise point (p₃ in the example) for an estimated milestone threshold.As shown, a path at a particular exercise point may represent a truepositive (TP) leading to exercising the option and a resulting benefit,or a false positive (FP) leading to exercising the option but aresulting regret. As other alternatives, a path at a particular exercisepoint may represent a true negative (TN) leading a decision to notexercise the option and a resulting averted failure, or a false negative(FN) leading to a decision to not exercise the option and a resultingomission, both of which may be generally referred to as a negative (N)since in either instance the decision may be made to forego exercisingthe option.

FIG. 9 furthers the example by illustrating a distribution of netconditional payoff values for a number of calculated net conditionalpayoff values (conditioned on a milestone threshold at a precedingexercise point P*p_(N−2)). Similar to FIG. 6, FIG. 9 includes aprobability accounting for those instances in which the conditionalpayoff has been limited to a predefined value, such as zero, to accountfor situations in which the exercise of the contingent claim wouldotherwise create a loss (negative) since a reasonably prudentparticipant will forego exercising the contingent claim in thesesituations.

As before, the milestone threshold P*p_(n) is intended to result in anet conditional payoff value of approximately zero, as again shown inblock 18 f. Thus, after determining the mean net conditional payoffvalue, if the mean net conditional payoff value does not equalapproximately zero, another milestone threshold P*p_(N−2) may beestimated for the preceding exercise point. The method may then repeatdetermining conditional distributions of contingent future value at theexpiration exercise point S_(T|P*p) _(N−2) and next-to-last exercisepoint S_(p) _(N−1) _(|P*p) _(N−2) , determining a conditional payoffS_(T|P*p) _(N−2) Payoff, net conditional payoff Net S_(T|P*p) _(N−2)Payoff and mean net conditional payoff Mean Net S_(T|P*p) _(N−2) Payoff,and determining if the mean net conditional payoff value equalsapproximately zero. The method may continue in this manner until anestimated milestone threshold P*p_(N−2) results in a mean netconditional payoff value equal to approximately zero, which may beselected as the milestone threshold at the respective exercise point.

Written in other terms, it may be said that the threshold milestoneP*p_(N−2) at the preceding exercise point p_(N−2) solves the followingexpression:E _(S) _(pN−2) └E _(S) _(pN−1) └Z ₀ ^(T)(r ₁ ,r ₂)|_((s) _(pN−2) _(=P*p)_(N−2) ₎ ┘−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ┘⁺ ×p(s _(p) _(N−1) ≧P*p_(N−1))≈x _(p) _(N−2) e ^(−r) ² ^(p) ^(N−2)The above expression may be interpreted as solving for the case wherethe difference between the discounted payoff at the expiration exercisepoint (t=T) and the estimated milestone threshold at the next-to-lastexercise point, conditioned on the asset value at the preceding exercisepoint equaling the estimated milestone threshold for that exercisepoint, and multiplied by the probability that asset value at thenext-to-last exercise point is greater than the milestone threshold forthat exercise point, approximately equals the discounted exercise priceat the preceding exercise point.

If or once the estimated milestone threshold at the preceding exercisepoint P*p_(N−2) results in a mean net conditional payoff value equal toapproximately zero, the method may similarly continue for each furtherpreceding exercise point, as again shown in block 18 g. Thus, and moregenerally for each exercise point p_(n), n=1, 2, . . . N−1, a milestonethreshold may be estimated P*p_(n); and conditional distributions ofcontingent future value may be determined for the expiration exercisepoint (distribution of contingent future benefits) S_(T|P*p) _(n) andany exercise points between the expiration exercise point and therespective exercise point S_(p) _(m) _(|P*p) _(n) , where m=n+1, n+2, .. . N−1; and p_(m)=p_(n+1), p_(n+2), . . . p_(N−1) (noting that(n+1)>(N−1) results in the empty sets m=Ø, and p_(m)=Ø). A conditionalpayoff at the respective exercise point S_(T|P*p) _(n) Payoff may bedetermined or otherwise calculated, such as in accordance with the DMalgorithm.

More particularly, for example, a conditional payoff (or intermediatepayoff) may be determined as the expected value of the differencebetween a present value conditional distribution of contingent futurebenefits at the expiration exercise point (discounted by r₁), and thepresent value of the exercise price (discounted by r₂) at the expirationexercise point, including limiting the minimum permissible difference toa predefined value, such as zero. And for each exercise point precedingthe next-to-last exercise point p_(N−1), determining the conditionalpayoff may further include reducing the expected value of the difference(intermediate payoff) by the present values of the exercise prices(discounted by r₂) at the one or more exercise points between theexpiration exercise point and the respective exercise point. Asindicated above, however, determining the conditional payoff may befurther conditioned on the conditional asset value at one or more, ifnot all, subsequent exercise points S_(p) _(m) _(|P*p) _(n) being atleast as much as (i.e., ≧) the estimated milestone threshold at therespective exercise points P*p_(m), such as by setting the conditionalpayoff to zero when the asset value at a subsequent exercise point isless than the milestone threshold at the respective subsequent exercisepoint. Thus, written notationally, the conditional payoff may bedetermined or otherwise calculated in accordance with the following:

$\begin{matrix}\begin{matrix}{{{IF}\mspace{14mu}{\forall_{m}\left( {S_{p_{m}|{P^{*}p_{n}}} \geq {P^{*}p_{m}}} \right)}},{then}} \\{{{S_{T|{P^{*}p_{n}}}{Payoff}} = {{E\left\lbrack {\max\left( {{{S_{T|{P^{*}p_{n}}}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack} - {\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}}}};} \\{{else},} \\{{S_{T|{P^{*}p_{n}}}{Payoff}} = 0}\end{matrix} & (12)\end{matrix}$And as a function of conditional asset values s_(p) _(m) _(|P*p) _(n) ,∀_(m) and s_(T|P*p) _(n) from respective conditional distributions S_(p)_(m) _(|P*p) _(n) , ∀_(m) and S_(T|P*p) _(n) , the conditional payoffmay be determined in accordance with the following:

$\begin{matrix}\begin{matrix}{{{IF}\mspace{14mu}{\forall_{m}\left( {s_{p_{m}|{P^{*}p_{n}}} \geq {P^{*}p_{m}}} \right)}},{then}} \\{{{S_{T|{P^{*}p_{n}}}{Payoff}} = {{\max\left( {{{s_{T|{P^{*}p_{n}}}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} - {\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}}}};} \\{{else},} \\{{S_{T|{P^{*}p_{n}}}{Payoff}} = 0}\end{matrix} & \left( {12a} \right)\end{matrix}$

As or after determining the conditional payoff for an exercise pointp_(n), a net conditional payoff at the respective exercise point may bedetermined or otherwise calculated, such as by reducing the conditionalpayoff by the present value of the exercise price (discounted by r₂) atthe respective exercise point. Written notationally, the net conditionalpayoff may be determined or otherwise calculated in accordance with thefollowing:NetS _(T|P*p) _(n) Payoff=S _(T|P*p) _(n)Payoff−x _(p) _(n) e ^(−r) ²^(p) ^(n)   (13)where x_(p) _(n) represents the exercise price (e.g., contingent futureinvestment) at the respective exercise point p_(n).

Then, from the net conditional payoff Net S_(T|P*p) _(n) Payoff, themean net conditional payoff Mean Net S_(T|P*p) _(n) Payoff at therespective exercise point may be determined by selecting or otherwiseforecasting a number of conditional asset values at the expirationexercise point (conditional future benefits) p_(N) (t=T) and anyexercise points between the expiration exercise point and the respectiveexercise point s_(p) _(m) _(|P*p) _(n) , ∀_(m) and s_(T|P*p) _(n) fromrespective conditional distributions of contingent future valueS_(T|P*p) _(n) and S_(p) _(m) _(|P*p) _(n) , ∀_(m); calculating, forthose forecasted conditional asset values, conditional payoff and netconditional payoff values such as in accordance with equations (12a) and(13); and calculating or otherwise determining the mean of thecalculated net conditional payoff values, such as in a manner similar tothat explained above. And if the mean net conditional payoff at therespective exercise point does not equal approximately zero, anothermilestone threshold P*p_(n) may be estimated for the respective exercisepoint, and the method repeated for determining a new mean netconditional payoff. In yet a further similar manner, the aforementionedsteps may be performed to effectuate equations (12) and (13) tocalculate P*p_(n), including the expected value expression of equation(12).

In other terms, it may be said that the threshold milestone P*p_(n) atexercise point p_(n) solves the following expression:E[E _(S) _(pn) . . . [E _(S) _(pN−1) [Z ₀ ^(T)(r ₁ ,r ₂)|_((s) _(pn)_(=P*p) _(n) ₎ ]−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ]⁺ . . . −x _(p)_(n) e ^(−r) ² ^(p) ^(n) ]⁺ ×p(s _(p) _(m) ≧P*p _(m), ∀_(m))≈x _(p) _(n)e ^(−r) ² ^(p) ^(n)The above expression may be interpreted by understanding that the valueof the multi-stage option may be viewed as the value of a series ofembedded expectations, with each expectation stage having an associatedcost to continue to the subsequent stage. These expectations may arisebecause of the uncertainty of the payoff at each stage, thus valued as amean or expected value of the sum total of all potential outcomes ateach stage. The expectations may be embedded because success (or netpayoff greater than zero) at an earlier stage allows forward progress tothe subsequent stage of the project. Failure (or net payoffs which arelimited to zero) may terminate forward progress of the project.Milestone thresholds P*p_(n) may be calculated for each stage orexercise point before the expiration exercise point. If project value ata subsequent stage or exercise point m is such that s_(p) _(m) ≧P*p_(m),there may be sufficient probability to expect that the subsequentstage(s) will be successful, and therefore the participant may incur acost x_(p) _(n) as the cost of proceeding to the next stage n+1. Thecost x_(p) _(n) may be weighted by the probability of actually incurringthe cost dependent on the success or failure of proceeding stages.

2. Arc Technique for Determining Milestone Thresholds

Referring now to FIG. 2 b, determining each milestone thresholdaccording to the arc technique of one exemplary embodiment of thepresent invention may include identifying J candidate milestonethresholds for the next-to-last exercise point P*p_(N−1,j), j=1, 2, . .. J, as shown in block 18 h. The candidate milestone thresholds for thisnext-to-last exercise point may be identified in any of a number ofdifferent manners. For example, the milestone thresholds for thenext-to-last exercise point may be identified to be a number ofequidistant values on either side of the determined mean asset value atthat exercise point μ_(S) _(pN−1) (see equation (1)).

Irrespective of exactly how the candidate milestone thresholds at thenext-to-last exercise point P*p_(N−1,j) are identified, respectivepayoffs or profits may be determined or otherwise calculated basedthereon, as shown in block 18 i. In this regard, the payoffs or profitsmay be conditioned on respective candidate milestone thresholds, anddetermined in any of a number of different manners, including inaccordance with the DM algorithm. Written notationally, for example, thepayoffs S_(T) Payoff|P*p_(N−1,j) may be determined as follows:

IF S_(p) _(N−1) ≧P*p_(N−1,j), thenS _(T)Payoff|P*p _(N−1,j) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0)];else,S _(T)Payoff|P*p _(N−1,j)=0  (14)And as a function of asset values s_(p) _(N−1) and s_(T) from respectivedistributions S_(p) _(N−1) and S_(T), the payoffs S_(T)Payoff|P*p_(N−1,j) may be determined as follows:

IF s_(p) _(N−1) ≧P*p_(N−1,j), thenS _(T)Payoff|P*p _(N−1,j)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0);else,S _(T)Payoff|P*p _(N−1,j)=0  (14a)

As or after determining the payoffs at the expiration exercise pointp_(N) (t=T), net payoffs may be calculated or otherwise determined byaccounting for the exercise price at the next-to-last exercise pointp_(N−1) (e.g., t=3), as shown in block 18 j. Similar to the payoffs, thenet payoffs can be determined in any of a number of different manners.In one embodiment, for example, the net payoffs may be determined bydiscounting the exercise price at the next-to-last exercise point by thesecond discount rate r₂ (e.g., risk-free rate), and subtracting thatdiscounted exercise price from respective payoffs at t=T. Writtennotationally, for example, the net payoffs Net S_(T) Payoff|P*p_(N−1,j)may be determined as follows:NetS _(T)Payoff|P*p _(N−1,j) =S _(T)Payoff|P*p _(N−1,j) −x _(p) _(N−1) e^(−r) ² ^(p) ^(N−1)   (15)where x_(p) _(N−1) represents the exercise price (e.g., contingentfuture investment) at the next-to-last exercise point p_(N−1) (e.g.,x_(p) _(N−1) =$30.00).

From the net payoffs, respective mean net payoffs Mean Net S_(T)Payoff|P*p_(N−1,j) may be determined for respective candidate milestonethresholds P*p_(N−1,j), as shown in block 18 k. For example, the meannet payoffs may be determined by first selecting or otherwiseforecasting a number of asset values s_(p) _(N−1) and s_(T) fromrespective distributions of contingent future value S_(p) _(N−1) andS_(T). For those forecasted asset values, then, respective payoff andnet payoff values may be calculated such as in accordance with equations(14a) and (15). The means of the calculated net payoff values may thenbe calculated or otherwise determined for respective candidate milestonethresholds P*p_(N−1,j). The aforementioned steps thereby effectuatingequations (14) and (15), including the expected value expression ofequation (14).

As indicated above, the milestone threshold P*p_(n) is intended tomaximize benefits (TP) and minimize regrets (FP) and omissions FN on arisk-adjusted basis. Thus, after determining the mean net payoff values,a maximum mean net payoff value may be selected from the determined orotherwise calculated mean net payoffs. The candidate milestone thresholdassociated with the respective mean net payoff value may then beselected as the milestone threshold at the respective exercise point, asshown in block 18 l. And in this regard, see FIG. 10 a for a graphplotting a number of mean net payoff values for a number of candidatemilestone thresholds, and including a selected milestone thresholdassociated with a maximum mean net payoff value.

As shown in block 18 m, after selecting a candidate milestone thresholdP*p_(N−1,j) that results in a maximum mean net payoff value, a number ofcandidate milestone thresholds may be identified for any exercise pointspreceding the next-to-last exercise point, such as by starting with theexercise point immediately preceding the next-to-last exercise pointP*p_(N−2), as again shown in block 18 h. As with the candidate milestonethresholds at the next-to-last exercise point, the candidate milestonethresholds for the preceding exercise point may be identified in any ofa number of different manners. Again, for example, the candidatemilestone thresholds for the preceding exercise point may be identifiedto be a number of equidistant values on either side of the determinedmean asset value at that exercise point, as explained above.

After identifying the candidate milestone thresholds at the precedingexercise point P*p_(N−2,j), respective payoffs or profits may bedetermined or otherwise calculated based thereon, as again shown inblock 18 i. In this regard, the payoffs or profits may be conditioned onrespective candidate milestone thresholds, and determined in any of anumber of different manners, including in accordance with the DMalgorithm. Written notationally, for example, the payoffs S_(T)Payoff|P*p_(N−2,j) may be determined as follows:

IF S_(p) _(N−2) ≧P*p_(N−2,j), S_(p) _(N−1) ≧P*p_(N−1), thenS _(T)Payoff|P*p _(N−2,j) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0)]−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ;else,S _(T)Payoff|P*p _(N−2,j)=0  (16)And as a function of asset values s_(p) _(N−2) , s_(p) _(N−1) and s_(T)from respective distributions S_(p) _(N−2) , S_(p) _(N−1) and S_(T), thepayoffs S_(T) Payoff|P*p_(N−2,j) may be determined as follows:

IF s_(p) _(N−2) ≧P*p_(N−2,j), s_(p) _(N−1) ≧P*p_(N−1), thenS _(T)Payoff|P*p _(N−2,j)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0)−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ;else,S _(T)Payoff|P*p _(N−2,j)=0  (16a)

As or after determining the payoffs at the expiration exercise pointp_(N) (t=T), net payoffs may be calculated or otherwise determined byaccounting for the exercise price at the preceding exercise pointp_(N−2) (e.g., t=1), as again shown in block 18 j. Similar to thepayoffs, the net payoffs can be determined in any of a number ofdifferent manners. In one embodiment, for example, the net payoffs maybe determined by discounting the exercise price at the precedingexercise point by the second discount rate r₂ (e.g., risk-free rate),and subtracting that discounted exercise price from respective payoffsat t=T. Written notationally, for example, the net payoffs Net S_(T)Payoff|P*p_(N−2,j) may be determined as follows:NetS _(T)Payoff|P*p _(N−2,j) =S _(T) Payoff|P*p _(N−2,j) −x _(p) _(N−2)e ^(−r) ² ^(p) ^(N−2)   (17)where x_(p) _(N−2) represents the exercise price (e.g., contingentfuture investment) at the preceding exercise point p_(N−2) (e.g., x_(p)_(N−2) =$15.00).

From the net payoffs, respective mean net payoffs Mean Net S_(T)Payoff|P*p_(N−2,j) may be determined for respective candidate milestonethresholds P*p_(N−2,j), as again shown in block 18 k. For example, themean net payoffs may be determined by first selecting or otherwiseforecasting a number of asset values from respective distributions ofcontingent future value S_(p) _(N−2) , S_(p) _(N−1) and S_(T). For thoseforecasted asset values, then, respective payoff and net payoff valuesmay be calculated such as in accordance with equations (16a) and (17).The means of the calculated net payoff values may then be calculated orotherwise determined for respective candidate milestone thresholdsP*p_(N−2,j). The aforementioned steps thereby effectuating equations(16) and (17), including the expected value expression of equation (16).

Similar to before, after determining the mean net payoff values, amaximum mean net payoff value may be selected from the determined orotherwise calculated mean net payoffs. The candidate milestone thresholdassociated with the respective mean net payoff value may then beselected as the milestone threshold for the respective exercise point,as again shown in block 18 l. And in this regard, see FIG. 10 b for agraph plotting a number of mean net payoff values for a number ofcandidate milestone thresholds, and including a selected milestonethreshold associated with a maximum mean net payoff value.

After selecting a candidate milestone threshold for the precedingexercise point P*p_(N−2) that results in a maximum mean net payoffvalue, the method may similarly continue for each further precedingexercise point, as again shown in block 18 m. Thus, and more generallyfor each exercise point p_(n), n=1, 2, . . . N−1, a number of candidatemilestone thresholds may be identified P*p_(n,j). Having selected thecandidate milestone thresholds P*p_(n,j), payoffs (or intermediatepayoffs) may then be determined as the expected value of the differencebetween a present value distribution of contingent future benefits atthe expiration exercise point (discounted by r₁), and the present valueof the exercise price (discounted by r₂) at the expiration exercisepoint, and including limiting the minimum permissible difference to apredefined value, such as zero. And for each exercise point precedingthe next-to-last exercise point p_(N−1), determining the payoffs mayfurther include reducing the expected values of the difference(intermediate payoffs) by the present values of the exercise prices(discounted by r₂) at the one or more exercise points between theexpiration exercise point and the respective exercise point. However,determining the payoffs may be further conditioned on the asset value atthe respective exercise point and one or more, if not all, subsequentexercise points before the expiration exercise point S_(p) _(n) being atleast as much as, if not greater than (i.e., ≧), the candidate milestonethreshold at the respective exercise points P*p_(n,j), such as bysetting the payoffs to zero when the asset value at an exercise point isless than the milestone threshold at the respective exercise point.Thus, written notationally, the payoffs may be determined or otherwisecalculated in accordance with the following:

$\begin{matrix}\begin{matrix}{{{{IF}\mspace{14mu} S_{p_{n}}} \geq {P^{*}p_{n,j}}},{\forall_{m}\left( {S_{p_{m}} \geq {P^{*}p_{m}}} \right)},{then}} \\{{\left. {S_{T}{Payoff}} \middle| {P^{*}p_{n,j}} \right. = {{E\left\lbrack {\max\left( {{{S_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack} - {\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}}}};} \\{{else},} \\{\left. {S_{T}{Payoff}} \middle| {P^{*}p_{n,j}} \right. = 0}\end{matrix} & (18)\end{matrix}$where m=n+1, n+2, . . . N−1; and p_(m)=p_(n+1), p_(n+2), . . . p_(N−1)(noting that (n+1)>(N−1) results in the empty sets m=Ø, and p_(m)=Ø).And as a function of asset values s_(p) _(n) , s_(p) _(m) , ∀_(m) ands_(T) from respective distributions S_(p) _(n) , S_(p) _(m) , ∀_(m) andS_(T), the payoffs S_(T) Payoff|P*p_(n,j) may be determined as follows:

$\begin{matrix}\begin{matrix}{{{{IF}\mspace{14mu} s_{p_{n}}} \geq {P^{*}p_{n,j}}},{\forall_{m}\left( {s_{p_{m}} \geq {P^{*}p_{m}}} \right)},{then}} \\{{\left. {S_{T}{Payoff}} \middle| {P^{*}p_{n,j}} \right. = {{\max\left( {{{s_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} - {\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}}}};} \\{{else},} \\{\left. {S_{T}{Payoff}} \middle| {P^{*}p_{n,j}} \right. = 0}\end{matrix} & \left( {18a} \right)\end{matrix}$

As or after determining the payoffs for an exercise point p_(n),respective net payoffs at the respective exercise point may bedetermined or otherwise calculated, such as by reducing the payoffs bythe present value of the exercise price (discounted by r₂) at therespective exercise point. Written notationally, the net payoffs may bedetermined or otherwise calculated in accordance with the following:NetS _(T)Payoff|P*p _(n,j) =S _(T)Payoff|P*p _(n,j) −x _(p) _(n) e ^(−r)² ^(p) ^(n)   (19)

From the net payoffs Net S_(T) Payoff|P*p_(n,j), respective mean netpayoffs Mean Net S_(T) Payoff|P*p_(n,j) may be determined for respectivecandidate milestone thresholds P*p_(n,j). Again, for example, the meannet payoffs may be determined by first selecting or otherwiseforecasting a number of asset values s_(p) _(n) , s_(p) _(m) , ∀_(m) ands_(T) from respective distributions of contingent future value S_(p)_(n) , S_(p) _(m) , ∀_(m) and S_(T). For those forecasted asset values,then, respective payoff and net payoff values may be calculated such asin accordance with equations (18a) and (19). The means of the calculatednet payoff values may then be calculated or otherwise determined, and amaximum mean net payoff value selected from the determined or otherwisecalculated mean net payoffs. The aforementioned steps being performed tothereby effectuate equations (18) and (19), including the expected valueexpression of equation (18). The candidate milestone thresholdassociated with the respective mean net payoff value may then beselected as the milestone threshold for the respective exercise pointp_(n).

3. Zero-Crossing Technique for Determining Milestone Thresholds

Referring to FIG. 2 c, determining each milestone threshold according tothe zero-crossing technique of one exemplary embodiment of the presentinvention may include selecting or otherwise forecasting K asset valuesat the next-to-last exercise point s_(p) _(N−1) _(,k), k=1, 2, . . . K,as shown in block 18 o. The asset values at this next-to-last exercisepoint may be selected or otherwise forecasted in any of a number ofdifferent manners. In one embodiment, for example, the asset values maybe selected or otherwise forecasted from the distribution of contingentfuture value S_(p) _(N−1) , such as in accordance with the Monte Carlotechnique for randomly generating values. In such instances, thedistribution of contingent future value may have been determined in themanner explained above with reference to equations (1) and (2).

Irrespective of exactly how the asset values are forecasted, respectivepayoffs or profits may be determined or otherwise calculated basedthereon, as shown in block 18 p. In this regard, the payoffs or profitsmay be conditioned on respective forecasted asset values, and determinedin any of a number of different manners, including in accordance withthe DM algorithm, such as in a manner similar to that explained abovefor the benefit-regret and arc techniques. Written notationally, thepayoffs S_(T) Payoff|s_(p) _(N−1) _(,k) may be determined as follows:S _(T)Payoff|s _(p) _(N−1) _(,k) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)]|s _(p) _(N−1) _(,k)  (20)And as a function of asset values s_(T) from the distribution ofcontingent future benefits S_(T), the payoffs S_(T) Payoff|s_(p) _(N−1)_(,k) may be determined as follows:S _(T)Payoff|s _(p) _(N−1) _(,k)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)|s _(p) _(N−1) _(,k)  (20a)

As or after determining the payoffs at the expiration exercise pointp_(N) (t=T), net payoffs may be calculated or otherwise determined byaccounting for the exercise price at the next-to-last exercise pointp_(N−1) (e.g., t=3), as shown in block 18 q. Similar to the payoffs, thenet payoffs can be determined in any of a number of different manners.In one embodiment, for example, the net payoffs may be determined bydiscounting the exercise price at the next-to-last exercise point by thesecond discount rate r₂ (e.g., risk-free rate), and subtracting thatdiscounted exercise price from respective payoffs at t=T. Writtennotationally, for example, the net payoffs Net S_(T) Payoff|s_(p) _(N−1)_(,k) may be determined as follows:NetS _(T)Payoff|s _(p) _(N−1) _(,k) =S _(T)Payoff|s _(p) _(N−1) _(,k) −x_(p) _(N−1) e ^(−r) ² ^(p) ^(N−1)   (21)where x_(p) _(N−1) represents the exercise price (e.g., contingentfuture investment) at the next-to-last exercise point p_(N−1) (e.g.,x_(p) _(N−1) =$30.00).

From the net payoffs, a positive root may be determined for a functiondefined based thereon ƒ(s_(p) _(N−1) )=Net S_(T) Payoff|s_(p) _(N−1) ,as shown in block 18 r. The function may comprise a function of any of anumber of different orders, and may be defined based upon the netpayoffs Net S_(T) Payoff|s_(p) _(N−1) _(,k) for respective forecastedasset values s_(p) _(N−1) _(,k) in any of a number of different manners.In one exemplary embodiment, example, the function may be defined byfirst selecting or otherwise forecasting a number of future benefitvalues s_(T,k) (asset values at the expiration exercise point) from thedistribution of contingent future benefits S_(T) based upon respectiveforecasted asset values s_(p) _(N−1) _(,k) and the correlationcoefficient Coeff_(p) _(N−1) _(,p) _(T) between the distribution ofcontingent future value at the next-to-last exercise point (i.e., S_(p)_(N−1) ) and the distribution of contingent future benefits(distribution of contingent future value) at the expiration exercisepoint (i.e., S_(T)) This operation may result in an array of pairedvalues └s_(p) _(N−1) _(,k),s_(T,k)┘. For those forecasted asset values,then, respective payoff and net payoff values may be calculated such asin accordance with equations (20a) and (21). The result of thisoperation, then, may be an array of paired values └s_(p) _(N−1)_(,k),Net S_(T) Payoff|s_(p) _(N−1) _(,k)┘.

After determining the net payoffs Net S_(T) Payoff|s_(p) _(N−1) _(,k)for respective forecasted asset values s_(p) _(N−1) _(,k), a functionmay be defined based on the respective forecasted asset values andassociated net payoffs. More particularly, for example, a second-orderquadratic function (or other order function) may be defined based on therespective values in accordance with a least-squares technique. In thisregard, see FIG. 11 a for a scatter plot of a number of net payoffvalues for a number of forecasted asset values, and including theexemplary quadratic function ƒ(s_(p) _(N−1) )=Net S_(T) Payoff|s_(p)_(N−1) =0.0008(s_(p) _(N−1) )²+0.1084(s_(p) _(N−1) )−29.983 definedbased thereon, again in the context of the example provided above.

Again, the milestone threshold P*p_(n) is intended to result in a finalvalue (payoff value) at the expiration exercise point substantiallyequal to the exercise price at the current exercise point and anysubsequent exercise points. Thus, after defining the function ƒ(s_(p)_(N−1) )=Net S_(T) Payoff|s_(p) _(N−1) , the function may be solved fora positive root thereof (i.e., value of s_(p) _(N−1) resulting in a netpayoff value Net S_(T) Payoff|s_(p) _(N−1) of approximately zero), suchas in accordance with any of a number of different techniques. Thisforecasted asset value may then be selected as the milestone thresholdfor the respective exercise point.

As shown in block 18 s, after finding a positive root of theaforementioned function to thereby determine the milestone thresholdP*p_(N−1), the technique may repeat for any exercise points precedingthe next-to-last exercise point. In this regard, K asset values may beselected or otherwise forecasted for any exercise points preceding thenext-to-last exercise point, such as by starting with the exercise pointimmediately preceding the next-to-last exercise point p_(N−2), as againshown in block 18 o. The asset values s_(p) _(N−2) _(,k) for thisnext-to-last exercise point may be selected or otherwise forecasted inany of a number of different manners. In one embodiment, for example,the asset values may be selected or otherwise forecasted from thedistribution of contingent future value S_(p) _(N−2) .

After selecting or otherwise forecasting the asset values, respectivepayoffs or profits may be determined or otherwise calculated basedthereon, as again shown in block 18 p. In this regard, the payoffs orprofits may be conditioned on respective forecasted asset values, anddetermined in any of a number of different manners, including inaccordance with the DM algorithm, such as in a manner similar to thatexplained above for the benefit-regret and arc techniques. Writtennotationally, the payoffs S_(T) Payoff|s_(p) _(N−2) _(,k) may bedetermined as follows:

IF S_(p) _(N−1) |s_(p) _(N−2) _(,k)≧P*p_(n−1), thenS _(T)Payoff|s _(p) _(N−2) _(,k) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)]|s _(p) _(N−2) _(,k) −x _(p) _(N−1) e ^(−r) ² ^(p)^(N−1) ;else,S _(T)Payoff|s _(p) _(N−2) _(,k)=0  (22)And as a function of asset values s_(p) _(N−1) and s_(T) from respectivedistributions of contingent future value S_(p) _(N−1) and S_(T), thepayoffs S_(T) Payoff|s_(p) _(N−2) _(,k) may be determined as follows:

IF s_(p) _(N−1) |s_(p) _(N−2) _(,k)≧P*p_(N−1), thenS _(T)Payoff|s _(p) _(N−2) _(,k)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)|s _(p) _(N−2) _(,k) −x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1);else,S _(T)Payoff|s _(p) _(N−2) _(,k)0  (22a)

As or after determining the payoffs at the expiration exercise pointp_(N) (t=T), net payoffs may be calculated or otherwise determined byaccounting for the exercise price at the preceding exercise pointp_(N−2) (e.g., t=1), as again shown in block 18 q. Similar to thepayoffs, the net payoffs can be determined in any of a number ofdifferent manners. In one embodiment, for example, the net payoffs maybe determined by discounting the exercise price at the precedingexercise point by the second discount rate r₂ (e.g., risk-free rate),and subtracting that discounted exercise price from respective payoffsat t=T. Written notationally, for example, the net payoffs Net S_(T)Payoff|s_(p) _(N−2) _(,k) may be determined as follows:NetS _(T)Payoff|s _(p) _(N−2) _(,k) =S _(T)Payoff|s _(p) _(N−2) _(,k) −x_(p) _(N−2) e ^(−r) ² ^(p) ^(N−2)   (23)

From the net payoffs, a positive root may be determined for a functiondefined based thereon ƒ(s_(p) _(N−2) )=Net S_(T) Payoff|s_(p) _(N−2) ,as again shown in block 18 r. The function may comprise a function ofany of a number of different orders, and may be defined based upon thenet payoffs Net S_(T) Payoff|s_(p) _(N−2) for respective forecastedasset values s_(p) _(N−2) _(,k) in any of a number of different manners.In this regard, the function may be defined by first selecting orotherwise forecasting a number of asset values at the next-to-lastexpiration point s_(p) _(N−1) _(,k) from a respective distribution ofcontingent future value S_(p) _(N−1) based upon the forecasted assetvalues s_(p) _(N−2) _(,k), and the correlation coefficient Coeff_(p)_(N−2) _(,p) _(N−1) . Similarly, for example, a number of asset valuesat the expiration exercise point s_(T,k) (future benefit values) may beselected or otherwise forecasted from the distribution of contingentfuture benefits (distribution of contingent future value) S_(T) basedupon respective forecasted asset values at the next-to-last expirationpoint p_(N−1), and the correlation coefficient Coeff_(p) _(N−1) _(,T).For those forecasted asset values, then, respective payoff and netpayoff values may be calculated such as in accordance with equations(22a) and (23).

After determining the net payoffs Net S_(T) Payoff|s_(p) _(N−2) _(,k)for respective forecasted asset values s_(p) _(N−2) _(,k), a functionmay be defined based thereon ƒ(s_(p) _(N−2) )=Net S_(T) Payoff|s_(p)_(N−2) , such as by defining a second-order, quadratic function (orother order function) based on the respective forecasted asset valuesand associated net payoffs in accordance with a least-squares technique.In this regard, see FIG. 11 b for a scatter plot of a number of netpayoff values for a number of forecasted asset values, and including theexemplary quadratic function ƒ(s_(p) _(N−2) )=Net S_(T) Payoff|s_(p)_(N−2) =0.002(s_(p) _(N−2) )²+0.054(s_(p) _(N−2) )−26.545 defined basedthereon, again in the context of the example provided above.

Again, the milestone threshold P*p_(n) is intended to result in a finalvalue (payoff value) at the expiration exercise point substantiallyequal to the exercise price at the current exercise point and anysubsequent exercise points. Thus, after a defining the function ƒ(s_(p)_(N−2) )=Net S_(T) Payoff|s_(p) _(N−2) , the function may be solved fora positive root thereof (i.e., value of s_(p) _(N−2) resulting in a netpayoff value Net S_(T) Payoff|s_(p) _(N−2) of approximately zero), suchas in accordance with any of a number of different techniques. Thisforecasted asset value may then be selected as the milestone thresholdfor the respective exercise point.

After finding a positive root of the aforementioned function to therebydetermine the milestone threshold P*p_(N−2), the technique may similarlycontinue for each further preceding exercise point, as again shown inblock 18 s. Thus, and more generally for each exercise point p_(n), n=1,2, . . . N−1, asset values s_(p) _(n) _(,k) may be selected or otherwiseforecasted from a respective distribution of contingent future valueS_(p) _(n) , such as in accordance with the Monte Carlo technique forrandomly generating values.

In addition to selecting or otherwise forecasting asset values s_(p)_(n) _(,k), payoffs (or intermediate payoffs) may be determined as theexpected value of the difference between a present value distribution ofcontingent future benefits at the expiration exercise point (discountedby r₁), and the present value of the exercise price (discounted by r₂)at the expiration exercise point, including limiting the minimumpermissible difference to a predefined value, such as zero. And for eachexercise point preceding the next-to-last exercise point p_(N−1),determining the payoffs may further include reducing the expected valuesof the difference (intermediate payoffs) by the present values of theexercise prices (discounted by r₂) at the one or more exercise pointsbetween the expiration exercise point and the respective exercise point.Again, however, determining the payoffs may be further conditioned onthe asset value at one or more, if not all, subsequent exercise pointsS_(p) _(m) |s_(p) _(n) _(,k) being at least as much as, if not greaterthan (i.e., ≧), the milestone threshold at the respective exercisepoints P*p_(m), such as by setting the payoffs to zero when the assetvalue for a subsequent exercise point is less than the milestonethreshold at the respective subsequent exercise point. Thus, writtennotationally, the payoffs may be determined or otherwise calculated inaccordance with the following:

$\begin{matrix}\begin{matrix}{{{IF}\mspace{14mu}{\forall_{m}\left( S_{p_{m}} \middle| {s_{p_{n},k} \geq {P^{*}p_{m}}} \right)}},{then}} \\{\left. {S_{T}{Payoff}} \middle| s_{p_{n,k}} \right. = \begin{matrix}{{{{E\left\lbrack {\max\left( {{{S_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack}}s_{p_{n},k}} -} \\{{\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}};}\end{matrix}} \\{{else},} \\{\left. {S_{T}{Payoff}} \middle| s_{p_{n}\;,k} \right. = 0}\end{matrix} & (24)\end{matrix}$And as a function of asset values s_(p) _(m) , ∀_(m) and s_(T) fromrespective distributions of contingent future value S_(p) _(m) , ∀_(m)and S_(T), the payoffs S_(T) Payoff|s_(p) _(n) _(,k) may be determinedas follows:

$\begin{matrix}\begin{matrix}{{{IF}\mspace{14mu}{\forall_{m}\left( s_{p_{m}} \middle| {s_{p_{n},k} \geq {P^{*}p_{m}}} \right)}},{then}} \\{\left. {S_{T}{Payoff}} \middle| s_{p_{n,k}} \right. = \begin{matrix}{{{{\max\left( {{{s_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)}}s_{p_{n},k}} -} \\{{\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}};}\end{matrix}} \\{{else},} \\{\left. {S_{T}{Payoff}} \middle| s_{p_{n}\;,k} \right. = 0}\end{matrix} & \left( {24a} \right)\end{matrix}$

As or after determining the payoffs for an exercise point p_(n),respective net payoffs at the respective exercise point may bedetermined or otherwise calculated, such as by reducing the payoffs bythe present value of the exercise price (discounted by r₂) at therespective exercise point. Written notationally, the net payoffs may bedetermined or otherwise calculated in accordance with the following:NetS _(T)Payoff|s _(p) _(n) _(,k) =S _(T)Payoff|s _(p) _(n) _(,k) −x_(p) _(n) e ^(−r) ² ^(p) ^(n)   (25)where x_(p) _(n) represents the exercise price (e.g., contingent futureinvestment) at the preceding exercise point p_(n).

From the net payoffs Net S_(T) Payoff|s_(p) _(n) _(,k), a positive rootmay be determined for a function ƒ(s_(p) _(n) )=Net S_(T) Payoff|s_(p)_(n) defined based on the net payoffs Net S_(T) Payoff|s_(p) _(n) _(,k)and respective forecasted asset values s_(p) _(n) _(,k). Again, forexample, the function may be defined by first selecting or otherwiseforecasting a number of asset values s_(p) _(m) _(,k), ∀_(m) fromrespective distributions of contingent future values S_(p) _(m) , ∀_(m)based upon the forecasted asset values s_(p) _(n) _(,k), and respectivecorrelation coefficients Coeff_(p) _(n) _(, p) _(m) , ∀_(m). Similarly,for example, a number of future benefit values s_(T,k) (asset values atthe expiration exercise point) may be selected or otherwise forecastedfrom the distribution of contingent future benefits S_(T) based uponrespective forecasted asset values s_(p) _(n) _(,k) and the correlationcoefficient Coeff_(p) _(n) _(,T). For those forecasted asset values,then, respective payoff and net payoff values may be calculated such asin accordance with equations (24a) and (25), after which a functionƒ(s_(p) _(n) )=Net S_(T) Payoff|s_(p) _(n) may be defined based on thenet payoffs Net S_(T) Payoff|s_(p) _(n) _(,k) and respective forecastedasset values s_(p) _(n) _(,k). This function may then be solved for itsroot, and that forecasted asset value may then be selected as themilestone threshold for the respective exercise point p_(n).

As explained above, K asset values s_(p) _(n) _(,k) may be forecasted(see step 18 o) before determining respective payoffs (see step 18 p),net payoffs (see step 18 q), and the root of the function ƒ(s_(p) _(n))=Net S_(T) Payoff|s_(p) _(n) (see step 18 r). This may be referred toas a static determination of the milestone threshold P*p_(n).Alternatively, however, the payoff, net payoffs, and root of theaforementioned function may be determined as each of the K asset valuesis forecast, in a manner referred to as a dynamic determination of themilestone threshold P*p_(n). In such instances, the payoff and netpayoffs may be determined based upon the currently forecasted assetvalue. The function and its root, however, may be determined based uponthe net payoff for the currently forecasted asset value, as well as thenet payoffs for any, or all, previously forecasted asset values. In thisregard, the function and its root may be built and refined as more assetvalues are forecasted.

4. Sorted List Technique for Determining Milestone Thresholds

Referring to FIG. 2 d, determining each milestone threshold according tothe sorted-list technique of one exemplary embodiment of the presentinvention may include selecting or otherwise forecasting K asset valuesat the next-to-last exercise point s_(p) _(N−1) _(,k), k=1, 2, . . . K,as shown in block 18 u. The asset values at this next-to-last exercisepoint may be selected or otherwise forecasted in any of a number ofdifferent manners. In one embodiment, for example, the asset values maybe selected or otherwise forecasted from the distribution of contingentfuture value S_(p) _(N−1) , such as in accordance with the Monte Carlotechnique for randomly generating values. In such instances, thedistribution of contingent future value may have been determined in themanner explained above with reference to equations (1) and (2).

Irrespective of exactly how the asset values are forecasted, respectivepayoffs or profits may be determined or otherwise calculated basedthereon, as shown in block 18 v. In this regard, the payoffs or profitsmay be conditioned on respective forecasted asset values, and determinedin any of a number of different manners, including in accordance withthe DM algorithm, such as in a manner similar to that explained abovefor the benefit-regret and arc techniques. Written notationally, thepayoffs S_(T) Payoff|s_(p) _(N−1) _(,k) may be determined as follows:

IF S_(p) _(N−1) ≧s_(p) _(N−1) _(,k), thenS _(T)Payoff|s _(p) _(N−1) _(,k) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)];else,S _(T)Payoff|s _(p) _(N−1) _(,k)=0  (26)And as a function of asset values s_(p) _(N−1) and s_(T) from respectivedistributions S_(p) _(N−1) and S_(T), the payoffs S_(T) Payoff|s_(p)_(N−1) _(,k) may be determined as follows:

IF s_(p) _(N−1) ≧s_(p) _(N−1) _(,k), thenS _(T)Payoff|s _(p) _(N−1) _(,k)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0);else,S _(T)Payoff|s _(p) _(N−1) _(,k)=0  (26a)

As or after determining the payoffs at the expiration exercise pointp_(N) (t=T), net payoffs may be calculated or otherwise determined byaccounting for the exercise price at the next-to-last exercise pointp_(N−1) (e.g., t=3), as shown in block 18 w. Similar to the payoffs, thenet payoffs can be determined in any of a number of different manners.In one embodiment, for example, the net payoffs may be determined bydiscounting the exercise price at the next-to-last exercise point by thesecond discount rate r₂ (e.g., risk-free rate), and subtracting thatdiscounted exercise price from respective payoffs at t=T. Writtennotationally, for example, the net payoffs Net S_(T) Payoff|P*p_(N−1,j)may be determined as follows:Net S _(T)Payoff|s _(p) _(N−1) _(,k) =S _(T) Payoff|s _(p) _(N−1) _(,k)−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1)   (27)

From the net payoffs, respective mean net payoffs Mean Net S_(T)Payoff|s_(p) _(N−1) _(,k) may be determined for respective forecastedasset values s_(p) _(N−1) _(,k), as shown in block 18 x. For example,the mean net payoffs may be determined by first selecting or otherwiseforecasting a number of asset values s_(p) _(N−1) and s_(T) fromrespective distributions of contingent future value S_(p) _(N−1) andS_(T). For those forecasted asset values, then, respective payoff andnet payoff values may be calculated such as in accordance with equations(26a) and (27). The means of the calculated net payoff values may thenbe calculated or otherwise determined for respective forecasted assetvalues s_(p) _(N−1) _(,k). The aforementioned steps thereby effectuatingequations (26) and (27), including the expected value expression ofequation (26).

As indicated above, the milestone threshold P*p_(n) is intended tomaximize benefits (TP) and minimize regrets (FP) and omissions FN on arisk-adjusted basis. Thus, after determining the mean net payoff values,a maximum mean net payoff value may be selected from the determined orotherwise calculated mean net payoffs. The forecasted asset value s_(p)_(N−1) _(,k) associated with the respective mean net payoff value maythen be selected as the milestone threshold P*p_(N−1) at the respectiveexercise point, as shown in block 18 y. And in this regard, see FIG. 12for a graph plotting a number of mean net payoff values for a number offorecasted asset values s_(p) _(N−1) _(,k), and including a selectedmilestone threshold associated with a maximum mean net payoff value.

As shown in block 18 z, after selecting a forecasted asset value s_(p)_(N−1) _(,k) that results in a maximum mean net payoff value to therebydetermine the milestone threshold P*p_(N−1), the technique may repeatfor any exercise points preceding the next-to-last exercise point. Inthis regard, K asset values may be selected or otherwise forecasted forany exercise points preceding the next-to-last exercise point, such asby starting with the exercise point immediately preceding thenext-to-last exercise point p_(N−2), as again shown in block 18 u. Theasset values s_(p) _(N−2) _(,k) for this next-to-last exercise point maybe selected or otherwise forecasted in any of a number of differentmanners. In one embodiment, for example, the asset values may beselected or otherwise forecasted from the distribution of contingentfuture value S_(p) _(N−2) .

After identifying the forecasted asset values at the preceding exercisepoint s_(p) _(N−2) _(,k), respective payoffs or profits may bedetermined or otherwise calculated based thereon, as again shown inblock 18 v. In this regard, the payoffs or profits may be conditioned onrespective forecasted asset values, and determined in any of a number ofdifferent manners, including in accordance with the DM algorithm.Written notationally, for example, the payoffs S_(T) Payoff|s_(p) _(N−2)_(,k) may be determined as follows:

IF S_(p) _(N−2) ≧s_(p) _(N−2) _(,k), S_(p) _(N−1) ≧P*p_(N−1), thenS _(T)Payoff|s _(p) _(N−2) _(,k) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)]−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ;else,S _(T)Payoff|s _(p) _(N−2) _(,k)=0  (28)And as a function of asset values s_(p) _(N−2) , s_(p) _(N−1) and s_(T)from respective distributions S_(p) _(N−2) , S_(p) _(N−1) and S_(T), thepayoffs S_(T) Payoff|s_(p) _(N−2) _(,k) may be determined as follows:

IF s_(p) _(N−2) ≧s_(p) _(N−2) _(,k), s_(p) _(N−1) ≧P*p_(N−1), thenS _(T)Payoff|s _(p) _(N−2) _(,k)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ;else,S _(T)Payoff|s _(p) _(N−2) _(,k)=0  (28a)

As or after determining the payoffs at the expiration exercise pointp_(N) (t=T), net payoffs may be calculated or otherwise determined byaccounting for the exercise price at the preceding exercise pointp_(N−2) (e.g., t=1), as again shown in block 18 w. Similar to thepayoffs, the net payoffs can be determined in any of a number ofdifferent manners. In one embodiment, for example, the net payoffs maybe determined by discounting the exercise price at the precedingexercise point by the second discount rate r₂ (e.g., risk-free rate),and subtracting that discounted exercise price from respective payoffsat t=T. Written notationally, for example, the net payoffs Net S_(T)Payoff|s_(p) _(N−2) _(,k) may be determined as follows:NetS _(T)Payoff|s _(p) _(N−2) _(,k) =S _(T)Payoff|s _(p) _(N−2) _(,k) −x_(p) _(N−2) e ^(−r) ² ^(p) ^(N−2)   (29)

From the net payoffs, respective mean net payoffs Mean Net S_(T)Payoff|s_(p) _(N−2) _(,k) may be determined for respective forecastedasset values s_(p) _(N−2) _(,k), as again shown in block 18 x. Forexample, the mean net payoffs may be determined by first selecting orotherwise forecasting a number of asset values from respectivedistributions of contingent future value S_(p) _(N−2) , S_(p) _(N−1) andS_(T). For those forecasted asset values, then, respective payoff andnet payoff values may be calculated such as in accordance with equations(28a) and (29). The means of the calculated net payoff values may thenbe calculated or otherwise determined for respective forecasted assetvalues s_(p) _(N−2) _(,k). The aforementioned steps thereby effectuatingequations (28) and (29), including the expected value expression ofequation (28).

Similar to before, after determining the mean net payoff values, amaximum mean net payoff value may be selected from the determined orotherwise calculated mean net payoffs. The forecasted asset valueassociated with the respective mean net payoff value may then beselected as the milestone threshold for the respective exercise point,as again shown in block 18 y.

After selecting a forecasted asset value for the preceding exercisepoint s_(p) _(N−2) _(,k) that results in a maximum mean net payoff valueto thereby determine the milestone threshold P*p_(N−2), the method maysimilarly continue for each further preceding exercise point, as againshown in block 18 z. Thus, and more generally for each exercise pointsp_(n), n=1, 2, . . . N−1, asset values s_(p) _(n) _(,k) may be selectedor otherwise forecasted from a respective distribution of contingentfuture value S_(p) _(n) , such as in accordance with the Monte Carlotechnique for randomly generating values.

In addition to selecting or otherwise forecasting asset values s_(p)_(n) _(,k), payoffs (or intermediate payoffs) may be determined as theexpected value of the difference between a present value distribution ofcontingent future benefits at the expiration exercise point (discountedby r₁), and the present value of the exercise price (discounted by r₂)at the expiration exercise point, including limiting the minimumpermissible difference to a predefined value, such as zero. And for eachexercise point preceding the next-to-last exercise point p_(N−1),determining the payoffs may further include reducing the expected valuesof the difference (intermediate payoffs) by the present values of theexercise prices (discounted by r₂) at the one or more exercise pointsbetween the expiration exercise point and the respective exercise point.However, determining the payoffs may be further conditioned on the assetvalue at the respective exercise point and one or more, if not all,subsequent exercise points before the expiration exercise point S_(p)_(n) being at least as much as, if not greater than (i.e., ≧), theforecasted asset values at the respective exercise points s_(p) _(n)_(,k), such as by setting the payoffs to zero when the asset value at anexercise point is less than the milestone threshold at the respectiveexercise point. Thus, written notationally, the payoffs may bedetermined or otherwise calculated in accordance with the following:

$\begin{matrix}\begin{matrix}{{{{IF}\mspace{14mu} S_{p_{n}}} \geq s_{p_{n},k}},{\forall_{m}\left( {S_{p_{m}} \geq {P^{*}p_{m}}} \right)},{then}} \\{{\left. {S_{T}{Payoff}} \middle| s_{p_{n},k} \right. = {{E\left\lbrack {\max\left( {{{S_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack} - {\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}}}};} \\{{else},} \\{\left. {S_{T}{Payoff}} \middle| s_{p_{n},k} \right. = 0}\end{matrix} & (30)\end{matrix}$where m=n+1, n+2, . . . N−1; and p_(m)=p_(n+1), p_(n+2), . . . p_(N−1)(noting that (n+1)>(N−1) results in the empty sets m=Ø, and p_(m)=Ø).And as a function of asset values s_(p) _(n) , s_(p) _(m) , ∀_(m) ands_(T) from respective distributions S_(p) _(n) , S_(p) _(m) , ∀_(m) andS_(T), the payoffs S_(T) Payoff|s_(p) _(n) _(,k) may be determined asfollows:

$\begin{matrix}\begin{matrix}{{{{IF}\mspace{14mu} s_{p_{n}}} \geq s_{p_{n},k}},{\forall_{m}\left( {s_{p_{m}} \geq {P^{*}p_{m}}} \right)},{then}} \\{{\left. {S_{T}{Payoff}} \middle| s_{p_{n},k} \right. = {{\max\left( {{{s_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} - {\sum\limits_{p_{m}}{x_{p_{m}}{\mathbb{e}}^{{- r_{2}}p_{m}}}}}};} \\{{else},} \\{\left. {S_{T}{Payoff}} \middle| s_{p_{n},k} \right. = 0}\end{matrix} & \left( {30a} \right)\end{matrix}$

As or after determining the payoffs for an exercise point p_(n),respective net payoffs at the respective exercise point may bedetermined or otherwise calculated, such as by reducing the payoffs bythe present value of the exercise price (discounted by r₂) at therespective exercise point. Written notationally, the net payoffs may bedetermined or otherwise calculated in accordance with the following:NetS _(T)Payoff|s _(p) _(n) _(,k) =S _(T)Payoff|s _(p) _(n) _(,k) −x_(p) _(n) e ^(−r) ² ^(p) ^(n)   (31)

From the net payoffs Net S_(T) Payoff|s_(p) _(n) _(,k), respective meannet payoffs Mean Net S_(T) Payoff|s_(p) _(n) _(,k) may be determined forrespective forecasted asset values s_(p) _(n) _(,k). Again, for example,the mean net payoffs may be determined by first selecting or otherwiseforecasting a number of asset values s_(p) _(n) , s_(p) _(m) , ∀_(m) ands_(T) from respective distributions of contingent future value S_(p)_(n) , S_(p) _(m) , ∀_(m) and S_(T). For those forecasted asset values,then, respective payoff and net payoff values may be calculated such asin accordance with equations (30a) and (31). The means of the calculatednet payoff values may then be calculated or otherwise determined, and amaximum mean net payoff value selected from the determined or otherwisecalculated mean net payoffs. The aforementioned steps being performed tothereby effectuate equations (30) and (31), including the expected valueexpression of equation (30). The forecasting asset value s_(p) _(n)_(,k), associated with the respective mean net payoff value may then beselected as the milestone threshold for the respective exercise pointp_(n).

Similar to the zero-crossing technique, K asset values s_(p) _(n) _(,k)may be forecasted (see step 18 u) before determining respective payoffs(see step 18 v), net and mean net payoffs (see steps 18 w and 18 x), andthe maximum mean net payoff Mean Net S_(T) Payoff|s_(p) _(n) (see step18 y). This may be referred to as a static determination of themilestone threshold P*p_(n). Alternatively, however, the payoff, net andmean net payoffs, and maximum mean net payoff may be determined as eachof the K asset values is forecast, in a manner referred to as a dynamicdetermination of the milestone threshold P*p_(n). In such instances, thepayoff, net and mean net payoffs may be determined based upon thecurrently forecasted asset value. The mean net payoff, however, may bedetermined based upon the net payoff for the currently forecasted assetvalue, as well as the net payoffs for any, or all, previously forecastedasset values. In this regard, the function and its root may be built andrefined as more asset values are forecasted.

Returning to FIG. 1, irrespective of exactly how the milestonethresholds P*p_(n) at the exercise points p_(n), n=1, 2, . . . N−1before the expiration exercise point p_(N) are estimated or otherwisedetermined, the value of the multi-stage option may thereafter bedetermined or otherwise calculated based thereon. The value of themulti-stage option may be determined in any of a number of differentmanners. For example, the value of the multi-stage option may beconsidered a payoff, which may be determined based upon the previouslydetermined distributions of contingent future value S_(p) _(n) , n=1, 2,. . . N, as shown in block 20. More particularly, for example,determining a payoff S_(T) Payoff may include determining the expectedvalue of the difference between a present value distribution ofcontingent future benefits at the expiration exercise point p_(N) (t=T)(discounted by r₁), and the present value of the exercise price(discounted by r₂) at the expiration exercise point, including limitingthe minimum permissible difference to a predefined value, such as zero.The expected value of the difference may be further reduced by thepresent values of the exercise prices (discounted by r₂) at the one ormore exercise points preceding the expiration exercise point (p_(n),n<N).

As indicated above, the milestone thresholds P*p_(n) at the exercisepoints may represent the minimum asset value (future benefits value) atwhich a reasonably prudent participant will exercise the contingentclaim at the respective exercise point. Thus, determining the payoff maybe further conditioned on the asset value at one or more, if not all,exercise points up to the expiration exercise point (p_(n+1), p_(n+2), .. . p_(N−1)) being at least as much as, if not greater than (i.e., ≧),the milestone threshold at the respective exercise points (P*p_(n+1),P*p_(n+2), . . . P*p_(N−1)), such as by setting or otherwise reducingthe payoff to zero when the asset value at an exercise point is lessthan the milestone threshold at the respective exercise point. Thus,written notationally, the payoff may be determined or otherwisecalculated in accordance with the following:

$\begin{matrix}\begin{matrix}{{{IF}\mspace{14mu}{\forall_{n < N}\left( {S_{p_{n}} \geq {P^{*}p_{n}}} \right)}},{then}} \\{{{S_{T}{Payoff}} = {{E\left\lbrack {\max\left( {{{S_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack} - {\sum\limits_{n < N}{x_{p_{n}}{\mathbb{e}}^{{- r_{2}}p_{n}}}}}};} \\{{else},} \\{{S_{T}{Payoff}} = 0}\end{matrix} & (32)\end{matrix}$And as a function of asset values s_(p) _(n) , ∀_(n<N) and s_(T) fromrespective distributions of contingent future value S_(p) _(n) , ∀_(n<N)and S_(T), the payoff S_(T) Payoff may be determined as follows:

$\begin{matrix}{{{{IF}\mspace{14mu}{\forall_{n < N}\left( {s_{p_{n}} \geq {P^{*}p_{n}}} \right)}},{then}}\mspace{20mu}{{{S_{T}{Payoff}} = {{\max\left( {{{s_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} - {\sum\limits_{n < N}{x_{p_{n}}{\mathbb{e}}^{{- r_{2}}p_{n}}}}}};}{{else},\text{}\mspace{20mu}{{S_{T}{Payoff}} = 0}}} & \left( {32a} \right)\end{matrix}$

As or after determining the payoff for the multi-stage option, the meanpayoff for the multi-stage option (value of the multi-stage option) maybe determined, such as by selecting or otherwise forecasting a number ofasset values s_(p) _(n) , ∀_(n<N) and s_(T) from respectivedistributions of contingent future value S_(p) _(n) , ∀_(n<N) and S_(T);calculating, for those forecasted asset values, payoff values such as inaccordance with equation (32a); and calculating or otherwise determiningthe mean of the calculated payoff values, such as in a manner similar tothat explained above, as shown in block 22. The aforementioned stepsthereby effectuating equation (32), including the expected valueexpression.

Written in other terms, the value of the multi-stage option may beexpressed as follows:E└E _(S) _(p1) . . . └E _(S) _(pN−2) └E _(S) _(pN−1) [Z ₀ ^(T)(r ₁ ,r₂)]−x _(p) _(N−1) e ^(−r) ² ^(p) ^(N−1) ┘−x _(p) _(N−1) e ^(−r) ² ^(p)^(N−2) ┘ . . . −x _(p) ₁ e ^(−r) ² ^(p) ¹ ┘×p(∀_(n<N)(s _(p) _(n) ≧P*p_(n)))The above expression may be interpreted as subtracting, from thediscounted payoff at the expiration exercise point (t=T), the discountedexercise prices for all of the exercise points before the expirationexercise point; and multiplying the resulting payoff by the probabilitythat, for the exercise points before the expiration exercise point, theasset values at the exercise points are greater than respectivemilestone thresholds for those exercise points. Also in the expressionthe “+” superscript represents a maximization function limiting thediscounted payoff to a minimum predefined value, such as zero.

From the above, the value of the multi-stage option can be understood asthe value of a series of embedded expectations, with each expectationstage having an associated cost to continue to the subsequent stage. Theexpectations may arise because of the uncertainty of the payoff at eachstage, thus valued as a mean or expected value of the sum total of allpotential outcomes at each stage. The expectations may be embeddedbecause success (or net payoff greater than zero) at an earlier stageallows forward progress to the subsequent stage of the project. Failure(or net payoffs which are limited to zero) may terminate forwardprogress of the project. Milestone thresholds P*p_(n) are calculated foreach stage or exercise point before the expiration exercise point. Ifproject value at Stage n is such that s_(p) _(n) >P*p_(n), there may besufficient probability to expect that the subsequent stage(s) will besuccessful, and therefore a cost x_(p) _(n) may be incurred to continueto the subsequent stage. The cost x_(p) _(n) may be weighted by theprobability of actually incurring the cost dependent on the success orfailure of proceeding stages. The value of the multi-stage option may bethe present value of the expiration point payoffs less the cost ofhaving traversed some or all the intermediate decision points, and thusincurred a cost x_(p) _(n) to continue to a subsequent point weighted bythe probability of actually traversing some or all of the intermediatedecision points.

FIG. 13 furthers the example of FIGS. 3, 4, 5 and 8, and illustrates anumber of different a conditional paths the asset value may take fromthe preceding exercise point, through the next-to-last exercise point tothe expiration exercise point for an initial mean asset value. Similarto FIG. 8, as shown in FIG. 13, a path at a particular exercise pointmay represent a TP, FP or N. FIG. 14, then, furthers the example byillustrating a distribution of payoff values for a number of calculatedpayoff values for valuation of the multi-stage option.

B. Early-Launch Option

In accordance with one exemplary embodiment of the present invention, acontingent-claim valuation may be performed for an early-launch optionover a period of time including one or more payout points (e.g.,dividend points), and an expiration point. A payout at a payout pointmay represent reduction or impairment of total value of the asset at theexpiration point. The impairment may be the result of any number ofevents, such as a competitor entering the market and taking market shareby sale of a competing product or technology, and thus reducing thetotal project value available at the expiration point. In suchinstances, in an effort to prevent the possible impairment of projectvalue, the product or technology may be launched early into the market,prior to that of the competitor's product or technology, thuspotentially preserving the entire project value. However, this mayrequire the flexibility to launch early, prior to the expectedexpiration point. Such additional flexibility may be valuable, and thevalue may be in addition to that of the expiration option. Valuation ofearly launch option in accordance with exemplary embodiments of thepresent invention therefore may calculate or otherwise determine thetotal value of the flexibility in determining a launch point thatattempts to preserve the full project value.

As indicated above, an early-launch option may include a contingentclaim at a single, variable exercise point, which may coincide with anyof the payout point(s) within the period of time. Exemplary embodimentsof the present invention will now be described with reference to aperiod of time including a single payout point at which the contingentclaim may or may not be exercised early, before the expiration point.But it should be understood that embodiments of the present inventionmay be equally applicable to instances in which the period of timeincludes multiple payout points.

Referring to FIG. 15, a method of performing a contingent claimvaluation of an early-launch option according to one exemplaryembodiment of the present invention may begin by defining a period oftime and a payout point within that period of time, as shown in block30. Again, the period of time can begin at t=0 and extend to t=T, andcan be divided into a number of different time segments. Within thedefined period of time, a time segment may correspond to the payoutpoint, and another time segment may correspond to a final or expirationexercise point at t=T, where the payout point and expiration exercisepoint may generally be referred to as “decision points” within theperiod of time. In one embodiment, for example, the time period T isdefined such that each time segment and decision point can berepresented as an integer divisors of T, i.e., t=0, 1, 2, . . . T. Thenumber N of decision points may be defined as p_(n), n=1, 2, . . . N≦T,where each p_(n) corresponds to a selected time segment of the period oftime; and where for one payout point, p₁ may refer to the payout point,and p₂ may refer to the expiration exercise point. Thus, for example,the period of time can be defined as a number of years (e.g., T=5)divided into a number of one-year time segments which, including theinitial time t=0, totals the number of years plus one time segment(e.g., t=0, 1, 2, . . . 5). Again, each time segment may begin at time tand end at time t+1 (presuming the time segment is an integer divisor ofT), and may be defined by the beginning time t. Thus, time segment t=1may extend from time t=1 to time t=2. Similarly, time segment t=2 mayextend from t=2 to t=3. For an example of the time segments for a periodof time, as well as the decision points within that period of time, seeTable 5 below.

TABLE 5 First Discount Rate 12% Second Discount Rate 5% Time Segment 0 12 3 4 5 Decision Point p₁ = 4 Expiration (p₂ = 5) Uncertainty 40% 40%40% 40% 40% 40% Payout Price $50.00 Exercise Price $140.00

Before, during or after defining the time period, a number of parametersmay be selected, determined or otherwise calculated for subsequent usein performing a contingent-claim valuation in accordance with thisexemplary embodiment of the present invention, as shown in block 32.These parameters may include first and second discount rates, r₁ and r₂;uncertainty, or volatility, in the market including the asset for eachdecision point p_(n) (or more generally for each segment of the periodof time t); a payout (e.g., dividend) price y that may impair the valueof the asset (future benefits value) subsequent to the payout point, andan exercise price x for the expiration exercise point. These parametersmay be selected, determined or otherwise calculated in any of a numberof different manners, such as in a manner similar to that explainedabove with reference to performing a multi-stage option valuation. Forexamples of these parameters, see Table 5. And more particularly as tothe payout price y, the price may be represented as an absolute value.Alternatively, however, the price y may be represented as a relativevalue with respect to a mean value of the asset at the respectivedecision point or at a subsequent decision point. For example, the pricey_(p) _(n) may be represented as a percentage of the asset value s_(p)_(n) at the respective decision point. Note that the payout price y_(p)_(n) may be defined as occurring immediately prior to S_(p) _(n) suchthat S_(p) _(n) =S_(p) _(n) −y_(p) _(n) , where S_(p) _(n) may representa distribution of contingent future value at the respective decisionpoint, but before the payout at that point.

Also before, during or after defining the time period, an initial, meanasset value may, but need not, be defined for the initial time segment(t=0), such as in a manner similar to that explained above withreference to performing a multi-stage option valuation, as shown inblock 34. Also, a revenue or value distribution S can be determined orotherwise calculated for each decision point p_(n) (payout andexpiration exercise points), and may be determined or otherwisecalculated along with correlation coefficients representingrelationships between successive distributions, as shown in block 36.Further, as suggested above, a revenue or value distribution may bedetermined for the payout point p₁, but before the payout; and also forthe payout point after the payout. Thus, p ₁ may refer to the payoutpoint before the payout at that point, and p₁ may refer to the payoutpoint after the payout. As before, each value distribution may beconsidered a distribution of contingent future value. In theearly-launch case, however, the distribution of contingent future valueat both the expiration exercise point and the decision point before theexpiration exercise point, p_(n), may be considered a distribution ofcontingent future benefits as the contingent claim may be exercised atthe expiration exercise point, or at a decision point before theexpiration exercise point. Each distribution of contingent futurebenefits S_(p) _(n) and correlation coefficient Coeff_(p) _(a) _(,p)_(b) can be determined in any of a number of different manners, such asin a manner similar to that explained above with reference to performinga multi-stage option valuation (including determining the mean valueμ_(S) _(pn) and standard deviation σ_(S) _(pn) ). As the payout price atthe payout point may impair the asset value subsequent to the payoutpoint, however, the mean value from which the distribution of contingentfuture benefits at the expiration exercise point may be determined in amanner that more particularly accounts for this impairment. Moreparticularly, for example, the mean value and standard deviation at theexpiration exercise point may be determined as follows:μ_(S) _(T) =(μ_(S) _(p1) −y _(p) ₁ )×e ^(r) ¹ ^((T−p) ¹ ⁾  (33)σ_(S) _(T) =μ_(S) _(T) ×√{square root over (e ^(u) ² ^(×(T−p) ¹⁾−1)}  (34)where μ_(S) _(p1) represents the mean value at the payout point and maybe determined in accordance with equation (1).

Continuing the example of Table 5 above, see Table 6 below for anexample of the initial mean value, as well as the mean values andstandard deviations each of the decision points p_(n), n=1, 2 (t=4, 5),and the correlation coefficient for the distribution at decision pointp₁ with reference to the distribution at decision point p₂. Further,FIG. 17 illustrates distributions of contingent future benefits S_(p)_(n) defined for decision points p_(n), n=1, 2 (t=4, 5) for the examplein Tables 1 and 2.

TABLE 6 Initial Mean Value $100.00 Time Segment 0 1 2 3 4 5 DecisionPoint p₁ = 4 Expiration (p₂ = 5) Mean Value $100.00 $161.61 $125.84Standard Deviation $0.00 $153.01 $139.31 Correlation Coefficient 0.89(p₄, p₅)

Irrespective of exactly how the distributions of contingent futurebenefits S_(p) _(n) are determined, the value of the early-launch optionmay be determined or otherwise calculated based thereon. Beforedetermining the value of the early-launch option, however, exemplaryembodiments of the present invention may account for situations in whicha reasonably prudent participant may exercise the option at the payoutpoint before the expiration exercise point, owing to a typical reductionin the expiration payoff due to the payout at the respective point. Moreparticularly, exemplary embodiments of the present invention maycalculate or otherwise determine a milestone threshold for the payoutpoint p₁, where the milestone threshold represents the minimum assetvalue at which a reasonably prudent participant will exercise thecontingent claim early at that payout point.

Similar to performing the contingent claim valuation of a multi-stageoption, performing a contingent claim valuation of an early-launchoption may include determining or otherwise calculating a milestonethreshold P* for the payout point, such as to facilitate maximizingbenefits (TP) and minimize regrets (FP) and omissions FN on arisk-adjusted basis, as shown in block 38. In this regard, the milestonethreshold for the payout point P*p₁ may correspond to the asset value atthe payout point likely to result in a final value at the expirationtime segment substantially equal to a value associated with exercisingthe option at the respective payout point (e.g., an asset value plusexpiration exercise price being expended at the payout point). Note thatthe milestone threshold P*p₁ may be defined with respect to the payoutpoint before the payout, such as with respect to S_(p) _(n) .

The milestone threshold for the payout point P*p₁ may be determined inany of a number of different manners, such as in accordance with one ormore of the aforementioned “benefit-regret” technique, “arc” technique,“zero-crossing” technique or “sorted list” technique for determining amilestone threshold, or in accordance with a “conditional” technique fordetermining a milestone threshold. The benefit-regret, arc, sorted listand conditional techniques will now be described below with reference toFIGS. 16 a-16 d.

1. Benefit-Regret Technique for Determining Milestone Thresholds

Referring to FIG. 16 a, determining the milestone threshold according tothe benefit-regret technique of one exemplary embodiment of the presentinvention may include estimating a milestone threshold for the payoutpoint P*p₁, as shown in block 38 a. The milestone threshold for thisnext-to-last decision point may be estimated in any of a number ofdifferent manners. For example, the milestone threshold for the payoutpoint may be estimated to be approximately equal to the determined meanasset value at that decision point μ_(S) _(p1) (see equation (1)).Continuing the example of Tables 5 and 6, see Table 7 below for a moreparticular example of an estimated milestone threshold for the payoutpoint p₁ (t=4) (the respective milestone threshold in the example beingrepresented by P*p₁=P*4).

TABLE 7 Time Segment 0 1 2 3 4 5 Decision Point p₁ = 1 Expiration (p₃ =5) Estimated Threshold (P*p₁) $200.00 S_(5|P*4) Mean Value $169.12S_(5|P*4) Standard Deviation $70.45

Irrespective of exactly how the milestone threshold for the payout pointP*p₁ is estimated, a value distribution at the expiration exercise pointp₂ (t=T) may thereafter be determined, where the value distribution isconditioned on the estimated milestone threshold at the payout point, asshown in block 38 b. In this regard, the value distribution may beconsidered a conditional distribution of contingent future benefits atthe expiration exercise point p₂ (t=T), conditioned on the estimatedvalue of the asset (milestone threshold) at the payout point P*p₁. Thisconditional distribution of contingent future benefits S_(T|P*p) ₁ maybe determined in any of a number of different manners (S_(T|P*p) _(n)more generally referring to the distribution of contingent futurebenefits, conditioned on the estimated value at the payout pointP*p_(n)). In one embodiment, for example, the conditional distributionof contingent future benefits may be determined based upon a conditionalmean asset value at the expiration exercise point μ_(S) _(T|P*p1) and aconditional standard deviation in time for the expiration exercise pointσ_(S) _(T|P*p1) , such as in accordance with the following:μ_(S) _(T|P*p1) =(P*p ₁ −y _(p) ₁ )×e ^(r) ¹ ^((T−p) ¹ ⁾  (35)σ_(S) _(T|P*p1) =μ_(S) _(T|P*p1) ×√{square root over (e ^(u) ² ^(×(T−p)¹ ⁾−1)}  (36)In the preceding equation (35), y_(p) ₁ represents the payout price atthe payout point p₁ (e.g., $50.00) (y_(p) _(n) more generallyrepresenting the payout price at the payout point p_(n)). For an exampleof the mean value and standard deviation for a conditional distributionof future benefits for the expiration exercise point p₂ (t=T) of theexample of Tables 5 and 6, and the estimated milestone threshold for thepayout point P*p_(N−1), see Table 7.

After determining the conditional mean and standard deviation for theexpiration exercise point, a conditional distribution of contingentfuture benefits at the expiration exercise point S_(T|P*p) ₁ can bedetermined by defining the conditional distribution according to therespective mean value and standard deviation. Again, the conditionaldistribution of contingent future benefits can be represented as any ofa number of different types of distributions but, in one embodiment, theconditional distribution of contingent future benefits is defined as alognormal distribution. In this regard, see FIG. 18 for a distributionof contingent future benefits S_(T) for the expiration exercise pointp₂, (t=5), along with an estimated milestone threshold P*p₁ andconditional distribution of contingent future benefits S_(T|P*p) ₁ forthe payout point p₁ (t=4), for the example of Tables 5, 6 and 7. Inaddition, FIG. 18 illustrates, with reference to the distribution ofcontingent future benefits, a similar distribution without accountingfor any impairment resulting from the payout of the payout price at thepayout point y_(p) ₁ .

Irrespective of exactly how the conditional distribution of contingentfuture benefits at the expiration exercise point S_(T|P*p) ₁ isdetermined, a conditional payoff or profit may be determined orotherwise calculated based thereon, as shown in block 38 c. Theconditional payoff S_(T|P*p) ₁ Payoff can be determined in any of anumber of different manners, such as in a manner similar to thatexplained above with reference to performing a multi-stage optionvaluation (see equations (6) and (6a)).

As or after determining the conditional payoff at the expirationexercise point p₂ (t=T), the net conditional payoff may be calculated orotherwise determined by accounting for the payout price at the payoutpoint p₁ (e.g., t=4) and again accounting for the exercise price at theexpiration exercise point, now representing a price for exercising theoption at the payout point, as shown in block 38 d. Similar to theconditional payoff, the net conditional payoff can be determined in anyof a number of different manners. In one embodiment, for example, thenet conditional payoff may be determined by discounting the estimatedmilestone threshold P*p₁ by the first discount rate and discounting theexercise price at the expiration exercise point by the second discountrate r₂ (accounting for early exercise of the contingent claim at thepayout point), and subtracting the difference of the discountedmilestone threshold and exercise price from the conditional payoff att=T. Written notationally, for example, the net conditional payoff NetS_(T|P*p) ₁ Payoff may be determined as follows:NetS _(T|P*p) ₁ Payoff=S _(T|P*p) ₁ Payoff−(P*p ₁ e ^(−r) ¹ ^(p) ¹ −x_(T) e ^(−r) ² ^(p) ¹ )  (37)where x_(T) again represents the exercise price (e.g., contingent futureinvestment) at the expiration exercise point p₂ (t=T) (e.g.,x_(T)=$140.00).

From the net conditional payoff, the mean net conditional payoff may bedetermined, as shown in block 38 e. For example, the mean netconditional payoff Mean Net S_(T|P*p) ₁ Payoff may be determined byselecting or otherwise forecasting a number of conditional asset valuess_(T|P*p) ₁ from the conditional distribution of contingent futurebenefits S_(T|P*p) ₁ ; calculating, for those forecasted asset values,conditional payoff and net conditional payoff values such as inaccordance with equations (6a) and (37); and calculating or otherwisedetermining the mean of the calculated net conditional payoff values.The aforementioned steps thereby effectuating equations (6) and (37),including the expected value expression of equation (6).

FIG. 19 continues the example of FIG. 18, and illustrates two of anumber of different a conditional paths the asset value may take fromthe payout point to the expiration exercise point for an estimatedthreshold, where one of the paths represents a true positive (TP)leading to a resulting benefit, and the other path represents a falsepositive (FP) leading to a resulting regret. And FIG. 20 furthers theexample by illustrating a distribution of net conditional payoff valuesfor a number of calculated net conditional payoff values (conditioned ona milestone threshold at the payout point P*p₁).

As indicated above, the milestone threshold P*p₁ is intended to resultin a final value (payoff value) at the expiration exercise pointsubstantially equal to a value associated with exercising the option atthe payout point (e.g., an asset value plus expiration exercise pricebeing expended at the payout point). In other words, the milestonethreshold is intended to result in a net conditional payoff value ofapproximately zero, as shown in FIG. 16 a, block 38 f, and in FIG. 20.Thus, after determining the mean net conditional payoff value Mean NetS_(T|P*p) ₁ Payoff, if the mean net conditional payoff value does notequal approximately zero, another milestone threshold P*p₁ may beestimated for the payout point P*p₁. The method may then repeatdetermining a conditional distribution of contingent future benefitsS_(T|P*p) ₁ , determining a conditional payoff S_(T|P*p) ₁ Payoff, netconditional payoff Net S_(T|P*p) ₁ Payoff and mean net conditionalpayoff Mean Net S_(T|P*p) ₁ Payoff, and determining if the mean netconditional payoff value equals approximately zero. The method maycontinue in this manner until an estimated milestone threshold P*p₁results in a mean net conditional payoff value equal to approximatelyzero. This estimated milestone threshold may then be considered themilestone threshold for the respective decision point. FIG. 21illustrates two of a number of different conditional paths the assetvalue may take from the payout point to the expiration exercise pointfor three different candidate milestone thresholds, again in the contextof the example provided above.

Written in other terms, it may be said that the threshold milestone P*p₁for the payout point p₁ solves the following expression:E[Z ₀ ^(T)(r ₁ ,r ₂)|_((s) _(p 1) _(=P*p1)) ]=E[Z ₀ ^(p) ¹ (r ₁ ,r₂)|_((s) _(p 1) _(=P*p1))],Z _(t) ₁ ^(t) ² (r ₁ ,r ₂)=[s _(t) ₂ e ^(−r) ¹ ^((t) ² ^(−t) ¹ ⁾ −x _(t)₂ e ^(−r) ² ^((t) ² ^(−t) ¹ ⁾]⁺,E[Z ₀ ^(p) ¹ (r ₁ ,r ₂)|_((s) _(p 1) _(=P*p1)) ]=[P*p ₁ e ^(−r) ¹ ^(p) ¹−x _(T) e ^(−r) ² ^(p) ¹ ]In the preceding, the milestone threshold P*p₁ may be the asset values_(p) ₁ of indifference between the decision to immediately launch atthe respective payout point before the payout in an attempt to preservetotal asset value, and the decision to wait or delay another period oftime until the expiration point, at which another decision to launch orterminate the project may be made. The valuation to immediately launchmay be equal to the asset value s_(p) ₁ less the exercise cost x_(T),and may be referred to as the intrinsic value. The valuation to wait ordelay may be subject to uncertainty and calculated according to asingle-stage option, such as in accordance with the DM algorithm, orZ-function. Thus, the expression s_(p) ₁ ≈P*p₁ may result when both theintrinsic value and the option value, discounted to t=0, are equal (orapproximately equal). Note that the intrinsic value defined at P*p₁ maybe a constant.

2. Arc Technique for Determining Milestone Thresholds

Referring to FIG. 16 b, determining each milestone threshold accordingto the arc technique of one exemplary embodiment of the presentinvention may include identifying J candidate milestone thresholds forthe payout point P*p_(1,j), j=1, 2, . . . J, as shown in block 38 h. Thecandidate milestone thresholds for the payout point may be identified inany of a number of different manners. For example, the milestonethresholds for the payout point may be identified to be a number ofequidistant values on either side of the determined mean asset value atthat point μ_(S) _(p1) (see equation (1)).

Irrespective of exactly how the candidate milestone thresholds for thepayout point P*p_(1,j) are identified, respective payoffs or profits maybe determined or otherwise calculated based thereon, as shown in block38 i. In this regard, the payoffs or profits may be conditioned onrespective candidate milestone thresholds, and determined in any of anumber of different manners, including in accordance with the DMalgorithm. Written notationally, for example, the payoffs S_(T)Payoff|P*p_(1,j) may be determined as follows:

IF S_(p) ₁ ≧P*p_(1,j), thenS _(T)Payoff|P*p _(1,j) =S _(p) ₁ e ^(−r) ¹ ^(p) ¹ −x _(T) e ^(−r) ²^(p) ¹ ;else,S _(T)Payoff|P*p _(1,j) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0)]  (38)And as a function of asset values s_(p) ₁ and s_(T) from respectivedistributions of contingent future value S_(p) ₁ and S_(T), the payoffsS_(T) Payoff|P*p_(1,j) may be determined as follows:

IF s_(p) ₁ ≧P*p_(1,j), thenS _(T)Payoff|P*p _(1,j) =s _(p) ₁ e ^(−r) ¹ ^(p) ¹ −x _(T) e ^(−r) ²^(p) ¹ ;else,S _(T)Payoff|P*p _(1,j)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0)  (38a)

As or after determining the payoffs at the expiration exercise point p₂(t=T), respective mean payoffs Mean S_(T) Payoff|P*p_(1,j) may bedetermined for respective candidate milestone thresholds P*p_(1,j), asshown in block 38 j. For example, the mean payoffs may be determined byfirst selecting or otherwise forecasting a number of asset values s_(p)₁ and s_(T) from respective distributions of contingent future benefitsS_(p) ₁ and S_(T). For those forecasted asset values, then, respectivepayoff values may be calculated such as in accordance with equation(38a). The means of the calculated payoff values may then be calculatedor otherwise determined for respective candidate milestone thresholdsP*p_(1,j). The aforementioned steps thereby effectuating equation (38),including its expected value expression.

Again, the milestone threshold P*p₁ is intended to maximize benefits(TP) and minimize regrets (FP) and omissions FN on a risk-adjustedbasis, as indicated above. Thus, after determining the mean payoffvalues, a maximum mean payoff value may be selected from the determinedor otherwise calculated mean payoffs. The candidate milestone thresholdassociated with the respective mean payoff value may then be selected asthe milestone threshold for the payout point, as shown in block 18 k.And in this regard, see FIG. 22 for a graph plotting a number of meannet payoff values for a number of candidate milestone thresholds, andincluding a selected milestone threshold associated with a maximum meanpayoff value.

3. Sorted List Technique for Determining Milestone Thresholds

Referring to FIG. 16 c, determining each milestone threshold accordingto the sorted-list technique of one exemplary embodiment of the presentinvention may include selecting or otherwise forecasting K asset valuesat the payout point s_(p) ₁ _(,k), k=1, 2, . . . K, as shown in block 38l. The asset values at the payout point may be selected or otherwiseforecasted in any of a number of different manners. In one embodiment,for example, the asset values may be selected or otherwise forecastedfrom the distribution of contingent future value S_(p) ₁ , such as inaccordance with the Monte Carlo technique for randomly generatingvalues. In such instances, the distribution of contingent future valuemay have been determined in the manner explained above with reference toequations (1) and (2).

Irrespective of exactly how the asset values are forecasted, respectivepayoffs or profits may be determined or otherwise calculated basedthereon, as shown in block 38 m. In this regard, the payoffs or profitsmay be conditioned on respective forecasted asset values, and determinedin any of a number of different manners, including in accordance withthe DM algorithm. Written notationally, for example, the payoffs S_(T)Payoff|s_(p) ₁ _(,k) may be determined as follows:

IF S_(p) ₁ ≧s_(p) ₁ _(,k), thenS _(T)Payoff|s _(p) ₁ _(,k) S _(p) ₁ e ^(−r) ¹ ^(p) ¹ −x _(T) e ^(−r) ²^(p) ¹ ;else,S _(T)Payoff|s _(p) ₁ _(,k) =E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r)² ^(T),0)]  (39)And as a function of asset values s_(p) ₁ and s_(T) from respectivedistributions of contingent future value S_(p) ₁ and S_(T), the payoffsS_(T) Payoff|s_(p) ₁ _(,k) may be determined as follows:

IF s_(p) ₁ ≧s_(p) ₁ _(,k), thenS _(T)Payoff|s _(p) ₁ _(,k) =s _(p) ₁ e ^(−r) ¹ ^(p) ¹ −x _(T) e ^(−r) ²^(p) ¹ ;else,S _(T)Payoff|s _(p) ₁ _(,k)=max(s _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0)  (39a)

As or after determining the payoffs at the expiration exercise point p₂(t=T), respective mean payoffs Mean S_(T) Payoff|s_(p) ₁ _(,k) may bedetermined for respective forecasted asset values s_(p) ₁ _(,k), asshown in block 38 n. For example, the mean payoffs may be determined byfirst selecting or otherwise forecasting a number of asset values s_(p)₁ and s_(T) from respective distributions of contingent future benefitsS_(p) ₁ and S_(T). For those forecasted asset values, then, respectivepayoff values may be calculated such as in accordance with equation(39a). The means of the calculated payoff values may then be calculatedor otherwise determined for respective forecasted asset values s_(p) ₁_(,k). The aforementioned steps thereby effectuating equation (39),including its expected value expression.

Similar to the case of the arc technique, the milestone threshold P*p₁is intended to maximize benefits (TP) and minimize regrets (FP) andomissions FN on a risk-adjusted basis, as indicated above. Thus, afterdetermining the mean payoff values, a maximum mean payoff value may beselected from the determined or otherwise calculated mean payoffs. Theforecasted asset value s_(p) ₁ _(,k) associated with the respective meanpayoff value may then be selected as the milestone threshold for thepayout point P*p₁, as shown in block 18 o.

4. Conditional Technique for Determining Milestone Thresholds

Referring to FIG. 16 d, determining the milestone threshold according tothe conditional technique of one exemplary embodiment of the presentinvention may include identifying J candidate milestone thresholds forthe payout point P*p_(1,j), j=1, 2, . . . J, as shown in block 38 p. Thecandidate milestone threshold for the payout point may be identified inany of a number of different manners. For example, the milestonethreshold for the payout point may be identified to be a number ofequidistant values on either side of the determined mean value of theasset at that payout point μ_(S) _(p1) (see equation (1)).

Irrespective of exactly how the candidate milestone thresholds for thepayout point P*p_(1,j) are identified, value distributions at theexpiration exercise point p₂ (t=T) may thereafter be determined, wherethe value distributions are conditioned on respective estimatedmilestone thresholds at the payout point, as shown in block 38 q. Inthis regard, the value distributions may be considered conditionaldistributions of contingent future benefits at the expiration exercisepoint, conditioned on respective asset values (candidate milestonethresholds) at the payout point P*p_(1,j). These conditionaldistributions of contingent future benefits S_(T|P*p) _(1,j) may bedetermined in any of a number of different manners. In one embodiment,for example, the conditional distributions of contingent future benefitsmay be determined based upon respective conditional mean asset values atthe expiration exercise point μ_(S) _(T|P*p1,j) and conditional standarddeviations in time at the expiration exercise point σ_(S) _(T|P*p1,j) ,such as in accordance with the following:μ_(S) _(T|P*p1,j) =(P*p _(1,j) −y _(p) ₁ )×e ^(r) ¹ ^((T−p) ¹ ⁾  (40)σ_(S) _(T|P*p1,j) =μ_(S) _(T|P*p1,j) ×√{square root over (e ^(u) ²^(×(T−p) ¹ ⁾−1)}  (41)

After determining the conditional means and standard deviations at theexpiration exercise point, the conditional distributions of contingentfuture benefits at the expiration exercise point S_(T|P*p) _(1,j) can bedetermined by defining the conditional distributions according to therespective means and standard deviations. Again, the conditionaldistributions of contingent future benefits can be represented as any ofa number of different types of distributions but, in one embodiment, theconditional distributions of contingent future benefits are defined aslognormal distributions.

After determining the conditional distribution of contingent futurebenefits S_(T|P*p) _(1,j) , respective conditional payoffs or profitsmay be determined or otherwise calculated based thereon, as shown inblock 38 r. The conditional payoffs S_(T|P*p) _(1,j) Payoff can bedetermined in any of a number of different manners, such as inaccordance with the DM algorithm. Written notationally, for example, theconditional payoffs may be determined as follows:S _(T|P*p) _(1,j) Payoff=E[max(S _(T|P*p) _(1,j) e ^(−r) ¹ ^(T) −x _(T)e ^(−r) ² ^(T),0)]  (42)And as a function of conditional asset values s_(T|P*p) _(1,j) from arespective distribution of contingent future value S_(T|P*p) _(1,j) ,the conditional payoffs S_(T|P*p) _(1,j) Payoff may be determined asfollows:S _(T|P*p) _(1,j) Payoff=max(s _(T|P*p) _(1,j) e ^(−r) ¹ ^(T) −x _(T) e^(−r) ² ^(T),0)  (42a)

Also after identifying the candidate milestone thresholds for the payoutpoint P*p_(1,j), respective payoffs or profits for exercising theearly-launch option at the payout p₁ may be determined or otherwisecalculated based thereon, as shown in block 38 s. These early exercisepayoffs P*p_(1,j) Payoff may be determined in any of a number ofdifferent manners, but in one exemplary embodiment, may be determined bydiscounting respective candidate milestone thresholds P*p_(1,j) by thefirst discount rate and discounting the exercise price at the expirationexercise point by the second discount rate r₂, and subtracting orotherwise differencing the discounted milestone threshold and exerciseprice. Written notationally, the early exercise payoffs may bedetermined as follows:P*p _(1,j)Payoff=P*p _(1,j) e ^(−r) ¹ ^(p) ¹ −x _(T) e ^(−r) ² ^(p) ¹  (43)

After determining the conditional payoffs S_(T|P*p) _(1,j) Payoff andthe early exercise payoffs P*p_(1,j) Payoff for the candidate milestonethresholds P*p_(1,j), the conditional payoffs may be compared torespective early exercise payoffs to identify a conditional payoffapproximately equal thereto. More particularly, for example, meanconditional payoffs may be compared to respective early exercise payoffsto identify a mean conditional payoff approximately equal thereto. Inthis regard, mean conditional payoffs may be determined by selecting orotherwise forecasting a number of conditional asset values s_(T|P*p)_(1,j) from a respective distribution of contingent future valueS_(T|P*p) _(1,j) ; calculating, for those forecasted conditional assetvalues, conditional payoff values such as in accordance with equation(42a); and calculating or otherwise determining the mean of thecalculated conditional payoff values. The aforementioned steps therebyeffectuating equation (42), including its expected value expression.

The situation where the mean conditional payoff approximately equals arespective early exercise payoff may represent a situation in which thefinal value or payoff foregoing early exercise of the early-launchoption approximately equals the final value or payoff having exercisedthe option at the payout point. And as such, the candidate milestonethreshold for which respective conditional and early exercise payoffsare approximately equal may be selected as the milestone threshold forthe respective decision point, as shown in block 38 t. In this regard,see FIG. 23 for a graph plotting conditional payoffs S_(T|P*p) _(1,j)Payoff and early exercise payoffs P*p_(1,j) Payoff for a number ofcandidate milestone thresholds P*p_(1,j), and identifying a milestonethreshold for which the conditional payoff and early exercise payoff areapproximately equal.

Irrespective of exactly how the milestone threshold P*p₁ for the payoutpoint is estimated or otherwise determined, the value of theearly-launch option may thereafter be determined or otherwise calculatedbased thereon. The value of the early-launch option may be determined inany of a number of different manners. The value of the early-launchoption may be considered a payoff S_(T) Payoff, which may represent thepayoff should the participant exercise the contingent claim at thepayout point, or the payoff should the participant forego early exerciseof the option and instead exercise the contingent claim at theexpiration exercise point. Again, the milestone threshold P*p₁ for thepayout point may therefore represent the minimum asset value (futurebenefits value) at which a reasonably prudent participant will exercisethe contingent claim early at that point. Thus, the particular payoffmay be determinable based upon the milestone threshold for the payoutpoint, as shown in block 40. Written notationally, for example, thepayoff S_(T) Payoff may be determined as follows:

IF S_(p) ₁ ≧P*p₁, thenS _(T)Payoff=S _(p) ₁ e ^(−r) ¹ ^(p) ¹ −x _(T) e ^(−r) ² ^(p) ¹ ;else,S _(T)Payoff=E[max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ²^(T),0)]  (44)And as a function of asset values s_(p) ₁ and s_(T) from respectivedistributions of contingent future value S_(p) ₁ and S_(T), the payoffS_(T) Payoff may be determined as follows:

IF s_(p) ₁ ≧P*p₁, thenS _(T)Payoff=s _(p) ₁ e ^(−r) ¹ ^(p) ¹ −x _(T) e ^(−r) ² ^(p) ¹ ;else,S _(T)Payoff=max(S _(T) e ^(−r) ¹ ^(T) −x _(T) e ^(−r) ² ^(T),0)  (44a)

As or after determining the payoff for the early-launch option, the meanpayoff for the early-launch option (value of the early-launch option)may be determined, such as by selecting or otherwise forecasting anumber of asset values s_(p) ₁ and s_(T) from respective distributionsof contingent future benefits S_(p) ₁ and S_(T); calculating, for thoseforecasted asset values, payoff values such as in accordance withequation (44a); and calculating or otherwise determining the mean of thecalculated payoff values, such as in a manner similar to that explainedabove, as shown in block 42. The aforementioned steps therebyeffectuating equation (44), including its expected value expression.

Written in other terms, the value of the early-launch option may beexpressed as follows:E[Z ₀ ^(T)(r ₁ ,r ₂)|_((s) _(p n) _(<P*p1)) ]×p(s _(p) ₁ <P*p ₁)+E[Z ₀^(p) ¹ (r ₁ ,r ₂)|_((s) _(p n) _(≧P*p1)) ]×p(s _(p) ₁ ≧P*p ₁),In the preceding, the value of the early-launch option may beinterpreted as the sum of the value of the opportunity to launch earlyto preserve the total project or asset value, and the value of thewaiting or delaying the launch decision until the expiration exercisepoint. At the expiration exercise point, additional information may beavailable about the impact of the impaired project value such that adecision may be made whether to launch or to terminate the project. Themilestone threshold P*p₁ may represent the value s_(p) ₁ at which thereasonably prudent participant may be indifferent between immediatelylaunching or exercising the option at the payout point (prior to thepayout) in an attempt to preserve total project or asset value, and thedecision to wait or delay until the expiration exercise point, at whichanother decision to launch or terminate the project may be made. At thismilestone threshold P*p₁, if the asset value is high such that s_(p) ₁≧P*p₁, then an early launch (exercising the contingent claim or option)may maximize the total value of the project. If, on the other hand, thevalue of the project is low such that s_(p) ₁ <P*p₁, then waiting ordelaying the decision to launch the project may maximize the total valueof the project or asset. The sum of the value of the opportunity tolaunch early, and the value of the waiting or delaying the launchdecision, may be weighted by the probability or likelihood of eachevent, thus appropriate valuing the early-launch option, reflecting thetotal value of flexibility.

FIG. 24 furthers the above example for the early-launch option, andillustrates a number of different a conditional paths the asset valuemay take from the beginning of the period of time, to the exemplarypayout point p₁, and if the payoff by early exercise of the option atthat decision point is less than or equal to the milestone threshold forthe respective point, through to the expiration exercise point. FIG. 25,then, furthers the example by illustrating a distribution of payoffvalues for a number of calculated payoff values for valuation of theearly-launch option.

C. Combination Option

In accordance with yet other exemplary embodiments of the presentinvention, a contingent-claim valuation may be performed for acombination option including a plurality of decision points over aperiod of time, where each decision point may include a go, no-godecision point and/or an early-launch decision point. Similar to thatexplained above, exemplary embodiments of the present invention will bedescribed with reference to a period of time including a plurality ofexercise points at which respective contingent claims may or may not beexercised (go, no-go decision points), and single payout point at whichthe contingent claim may or may not be exercised early, before theexpiration point (early-launch decision point). It should be understood,however, that exemplary embodiments of the present invention may beequally applicable to instances in which the period of time includes asingle go, no-go decision point and a plurality of early-launch decisionpoints, a plurality of go, no-go decision points and early-launchdecision points, or a single go, no-go decision point and a singleearly-launch decision point.

Referring to FIG. 26, a method of performing a contingent claimvaluation of a combination option according to one exemplary embodimentof the present invention may begin by defining a period of time, aplurality of exercise point and a payout point within that period oftime, as shown in block 50. Again, the period of time can begin at t=0and extend to t=T, and can be divided into a number of different timesegments. Within the defined period of time, a time segment maycorrespond to one of the exercise points and/or the payout point, andanother time segment may correspond to a final or expiration exercisepoint at t=T, where the exercise points (including the expirationexercise point) and payout point may generally be referred to as“decision points” within the period of time. In one embodiment, forexample, the time period T is defined such that each time segment anddecision point can be represented as an integer divisors of T, i.e.,t=0, 1, 2, . . . T. The number N of decision points may be defined asp_(n), n=1, 2, . . . N≦T. In the preceding, each p_(n) corresponds to aselected time segment of the period of time, and may include an exercisepoint ep_(n′) (n′=1, 2, . . . N′≦N) and/or a payout point pp_(n″) (n″=1,2, . . . N″≦N; and for one payout point n″=1). For an example of thetime segments for a period of time, as well as the decision pointswithin that period of time, see Table 8 below.

TABLE 8 First Discount Rate 12% Second Discount Rate 5% Time Segment 0 12 3 4 5 Decision Point  p₁ = 1  p₂ = 4 Expiration ep₁ = 1 ep₂ = 4  (p₃ =5) pp₁ = 4 (ep₃ = 5) Uncertainty 60% 60% 60% 60% 60% 40% Payout Price$50.00 Exercise Price $5.00 $10.00 $140.00

Before, during or after defining the time period, a number of parametersmay be selected, determined or otherwise calculated for subsequent usein performing a contingent-claim valuation in accordance with thisexemplary embodiment of the present invention, as shown in block 52.These parameters may include first and second discount rates, r₁ and r₂;uncertainty, or volatility, in the market including the asset for eachdecision point p_(n) (or more generally for each segment of the periodof time t); a payout (e.g., dividend) price y that may impair the valueof the asset (future benefits value) subsequent to the payout point, andexercise prices x for respective exercise points (including theexpiration exercise point). These parameters may be selected, determinedor otherwise calculated in any of a number of different manners, such asin a manner similar to that explained above with reference to performinga multi-stage option valuation or an early-launch option valuation. Forexamples of these parameters, see Table 8.

Also before, during or after defining the time period, an initial, meanasset value may, but need not, be defined for the initial time segment(t=0), such as in a manner similar to that explained above withreference to performing a multi-stage option valuation or early-launchoption valuation, as shown in block 54. Also, a revenue or valuedistribution S can be determined or otherwise calculated for eachdecision point p_(n), and may be determined or otherwise calculatedalong with correlation coefficients representing relationships betweensuccessive distributions, as shown in block 56. As before, each valuedistribution may be considered a distribution of contingent futurevalue. Each distribution of contingent future benefits S_(p) _(n) andcorrelation coefficient Coeff_(p) _(a) _(, p) _(b) can be determined inany of a number of different manners, such as in a manner similar tothat explained above with reference to performing a multi-stage optionvaluation or early-launch option (including determining the mean valueμ_(S) _(pn) and standard deviation σ_(S) _(pn) ). In various instances,however, the correlation coefficient may be modified or otherwiseadjusted from its calculated value, or one or more of the uncertaintiesupon which the correlation coefficient may be calculated may be modifiedor otherwise adjusted. In this regard, the payout at the payout pointmay introduce additional, instantaneous volatility or uncertainty, whichmay effectuate a reduction in the correlation coefficient in a mannerproportional to the size of the payout relative to the asset value atthe respective payout point.

Further, as the payout price at the payout point may impair the assetvalue subsequent to the payout point, however, the mean value from whichthe distribution of contingent future benefits at the first exercisepoint following the payout point (and, in turn, the remaining exercisepoints) may be determined in a manner that more particularly accountsfor this impairment. More particularly, for example, the mean value andstandard deviation at an exercise point p_(n+)1 subsequent a payoutpoint p_(n) may be determined as follows:μ_(S) _(n+1) =(μ_(S) _(pn) −y _(p) _(n) )×e ^(r) ¹ ^((p) ^(n+1) ^(−p) ²⁾  (45)σ_(S) _(n+1) =μ_(S) _(n+1) ×√{square root over (e ^(u) ² ^(×(p) ^(n+1)^(−p) ¹ ⁾−1)}  (46)where μ_(S) _(n) represents the mean value at the payout point and maybe determined in accordance with equation (1).

Continuing the example of Table 8 above, see Table 9 below for anexample of the initial mean value, as well as the mean values andstandard deviations each of the decision points p_(n), n=1, 2, 3 (t=1,4, 5), and the correlation coefficients Coeff_(p) ₁ _(, p) ₂ andCoeff_(p) ₂ _(, p) ₃ .

TABLE 9 Initial Mean Value $100.00 Time Segment 0 1 2 3 4 5 DecisionPoint p₁ = 1 p₂ = 4 Expiration (p₃ = 5) Mean Value $100.00 $112.75$161.61 $125.84 Standard Deviation $0.00 $74.22 $153.01 $139.31Correlation Coefficient 0.50 0.89 (p₁, p₂) (p₂, p₃)

Irrespective of exactly how the distributions of contingent futurebenefits S_(p) _(n) are determined, the value of the combination optionmay be determined or otherwise calculated based thereon. Beforedetermining the value of the combination option, however, exemplaryembodiments of the present invention may account for situations in whicha reasonably prudent participant may not exercise an option at aparticular exercise point (and thus any remaining exercise points); ormay exercise the option at the payout point before the expirationexercise point, owing to a typical reduction in the expiration payoffdue to the payout at the respective point. Similar to that above,exemplary embodiments of the present invention may calculate orotherwise determine milestone thresholds for the exercise and payoutpoints. More particularly, exemplary embodiments of the presentinvention may calculate or otherwise determine go, no-go milestonethresholds P*ep_(n′) for respective exercise points before theexpiration exercise point; and/or calculate or otherwise determine anearly-launch milestone threshold P*pp_(n″) for the payout point. Again,the go, no-go milestone threshold for each exercise point represents theminimum asset value at which a reasonably prudent participant willexercise the contingent claim at that exercise point; and theearly-launch milestone threshold for the payout point represents theminimum asset value at which a reasonably prudent participant willexercise the contingent claim early at that payout point.

Similar to performing the contingent claim valuation of a multi-stageoption or an early-launch option, performing a contingent claimvaluation of a combination option may include determining or otherwisecalculating milestone thresholds for the exercise points P*ep_(n′) andthe payout point P*pp_(n″), such as to facilitate maximizing benefits(TP) and minimize regrets (FP) and omissions FN on a risk-adjustedbasis, as shown in block 58. In this regard, the milestone threshold foreach exercise point P*ep_(n′) may correspond to the asset value at therespective exercise point likely to result in a risk-adjusted,discounted final value at the expiration time segment substantiallyequal to the risk-adjusted, discounted exercise price at the currentexercise point and any subsequent exercise points. And the milestonethreshold for the payout point P*pp_(n″) may correspond to the assetvalue at the payout point likely to result in a final value at theexpiration time segment substantially equal to a value associated withexercising the option at the respective payout point (e.g., an assetvalue plus expiration exercise price being expended at the payoutpoint).

The milestone thresholds for the exercise points P*ep_(n′) and thepayout point P*pp_(n″) may be determined in any of a number of differentmanners, such as in accordance with one or more of the aforementioned“benefit-regret” technique, “arc” technique, “zero-crossing” technique,“sorted list” technique or “conditional” technique for determining amilestone threshold, such as in a manner similar to that explained abovefor the multi-stage option and early-launch option. The benefit-regretand arc techniques will now be more particularly described below withreference to FIGS. 27 a and 27 b.

1. Benefit-Regret Technique for Determining Milestone Thresholds

Determining the milestone threshold for each exercise point P*ep_(n′)according to the benefit-regret technique of one exemplary embodiment ofthe present invention may proceed in a manner similar to that for themulti-stage option. Thus, and generally for each exercise point ep_(n′),n′=1, 2, . . . N′−1, a milestone threshold may be estimated P*ep_(n′)(see FIG. 2 a, block 18 a). Also, conditional distributions ofcontingent future value may be determined for the expiration exercisepoint (distribution of contingent future benefits) S_(T|P*ep) _(n′) andany exercise points between the expiration exercise point and therespective exercise point S_(p) _(m′) _(|P*ep) _(n′) , where m′=n′+1,n′+2, . . . N′−1; and ep_(m′)=ep_(n′+1), ep_(n′+2), . . . ep_(N′−1)(noting that (n′+1)>(N′−1) results in the empty sets m′=Ø, andep_(m′)=Ø) (see block 18 b). Continuing the example of Tables 8 and 9,see Table 10 below for a more particular example of an estimatedmilestone threshold at the next-to-last exercise point ep₂ (t=4) (therespective milestone threshold in the example being represented byP*ep₂=P*4′), as well as the mean and standard deviation for an exemplaryconditional distribution of contingent future benefits S_(T|P*ep) ₂ .

TABLE 10 Time Segment 0 1 2 3 4 5 Exercise Point ep₁ = 1 ep₂ = 4Expiration (ep₃ = 5) Estimated Threshold (P*4′) $170.00 S_(5\P*4′)MeanValue $135.30 S_(5|P*4′)Standard Deviation $89.06 Estimated Threshold(P*4″) $198.00 S_(5|P*4″)Mean Value $166.87 S_(5|P*4″)Standard Deviation$109.85

After estimating a milestone threshold and determining the conditionaldistributions of contingent future value, a conditional payoff at therespective exercise point S_(T|P*ep) _(n′) Payoff may be determined orotherwise calculated, such as in accordance with the DM algorithm (seeblock 18 c). More particularly, for example, a conditional payoff (orintermediate payoff) may be determined as the expected value of thedifference between a present value conditional distribution ofcontingent future benefits at the expiration exercise point (discountedby r₁), and the present value of the exercise price (discounted by r₂)at the expiration exercise point, including limiting the minimumpermissible difference to a predefined value, such as zero. And for eachexercise point preceding the next-to-last exercise point ep_(N′−1),determining the conditional payoff may further include reducing theexpected value of the difference (intermediate payoff) by the presentvalues of the exercise prices (discounted by r₂) at the one or moreexercise points between the expiration exercise point and the respectiveexercise point. As indicated above, however, determining the conditionalpayoff may be further conditioned on the conditional asset value at oneor more, if not all, subsequent exercise points S_(p) _(m′) _(|P*ep)_(n′) being at least as much as (i.e., ≧) the estimated milestonethreshold at the respective exercise points P*ep_(m′), such as bysetting the conditional payoff to zero when the asset value at asubsequent exercise point is less than the milestone threshold at therespective subsequent exercise point. Thus, written notationally, theconditional payoff may be determined or otherwise calculated inaccordance with the following:

$\begin{matrix}{{{{IF}\mspace{14mu}{\forall_{m^{\prime}}\left( {S_{p_{m^{\prime}}|{P^{*}{ep}_{n^{\prime}}}} \geq {P^{*}{ep}_{m^{\prime}}}} \right)}},{then}}\mspace{20mu}{{{S_{T|{P^{*}{ep}_{n^{\prime}}}}{Payoff}} = {{E\left\lbrack {\max\left( {{{S_{T|{P^{*}{ep}_{n^{\prime}}}}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack} - {\sum\limits_{{ep}_{m^{\prime}}}{x_{{ep}_{m^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{m^{\prime}}}}}}};}{{else},\text{}\mspace{20mu}{{S_{T|{P^{*}{ep}_{n^{\prime}}}}{Payoff}} = 0}}} & (47)\end{matrix}$And as a function of conditional asset values s_(ep) _(m′) _(|P*ep)_(n′) , ∀_(m′) and s_(T|P*ep) _(n′) from respective conditionaldistributions S_(ep) _(m′) _(|P*ep) _(n′) , ∀_(m′) and S_(T|P*ep) _(n′), the conditional payoff may be determined in accordance with thefollowing:

$\begin{matrix}{{{{IF}\mspace{14mu}{\forall_{m^{\prime}}\left( {s_{{ep}_{m^{\prime}}|{P^{*}{ep}_{n^{\prime}}}} \geq {P^{*}{ep}_{m^{\prime}}}} \right)}},{then}}\mspace{20mu}{{{S_{T|{P^{*}{ep}_{n^{\prime}}}}{Payoff}} = {{\max\left( {{{s_{T|{P^{*}{ep}_{n^{\prime}}}}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} - {\sum\limits_{{ep}_{m^{\prime}}}{x_{{ep}_{m^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{m^{\prime}}}}}}};}{{else},\text{}\mspace{20mu}{{S_{T|{P^{*}{ep}_{n^{\prime}}}}{Payoff}} = 0}}} & \left( {47a} \right)\end{matrix}$

As or after determining the conditional payoff for an exercise pointep_(n′), a net conditional payoff at the respective exercise point maybe determined or otherwise calculated, such as by reducing theconditional payoff by the present value of the exercise price(discounted by r₂) at the respective exercise point (see block 18 d).Written notationally, the net conditional payoff may be determined orotherwise calculated in accordance with the following:NetS _(T|P*ep) _(n′) Payoff=S _(T|P*ep) _(n′) Payoff−x _(ep) _(n′) e^(−r) ² ^(ep) ^(n′)   (48)where x_(ep) _(n′) represents the exercise price (e.g., contingentfuture investment) at the respective exercise point ep_(n′).

Then, from the net conditional payoff Net S_(T|P*ep) _(n′) Payoff, themean net conditional payoff Mean Net S_(T|P*ep) _(n′) Payoff at therespective exercise point may be determined, such as by selecting orotherwise forecasting a number of conditional asset values at theexpiration exercise point (conditional future benefits) ep_(N′) (t=T)and any exercise points between the expiration exercise point and therespective exercise point s_(ep) _(m′) _(|P*ep) _(n′) , ∀_(m′) ands_(T|P*ep) _(n′) from respective conditional distributions of contingentfuture value S_(T|P*ep) _(n′) and S_(ep) _(m′) _(|P*ep) _(n′) , ∀_(m′);calculating, for those forecasted conditional asset values, conditionalpayoff and net conditional payoff values such as in accordance withequations (47a) and (48); and calculating or otherwise determining themean of the calculated net conditional payoff values, such as in amanner similar to that explained above (see block 18 e). And if the meannet conditional payoff at the respective exercise point does not equalapproximately zero, another milestone threshold P*ep_(n′) may beestimated for the respective exercise point, and the method repeated fordetermining a new mean net conditional payoff (see block 18 f). In yet afurther similar manner, the aforementioned steps may be performed toeffectuate equations (47) and (48), including the expected valueexpression of equation (47).

Determining the milestone threshold for the payout point P*pp₁ accordingto the benefit-regret technique of one exemplary embodiment of thepresent invention may proceed in a manner similar to that for theearly-launch option. Thus, determining the milestone threshold mayinclude estimating a milestone threshold for the payout point P*pp₁ (seeFIG. 16 a, block 38 a). Again, see Table 10 above for a more particularexample of an estimated milestone threshold for the payout point pp₁(t=4) (the respective milestone threshold in the example beingrepresented by P*pp₁=P*4″). Then, a distribution of contingent futurebenefits at the expiration exercise point p₂ (t=T), conditioned on theestimated value of the asset (milestone threshold) at the payout pointP*pp₁, may be determined (see block 38 b). For an example of the meanvalue and standard deviation for an exemplary conditional distributionof future benefits S_(T|P*pp) ₁ , see Table 10.

The method of determining the milestone threshold for the payout pointmay further include determining or otherwise calculating a conditionalpayoff or profit (see block 38 c). The conditional payoff S_(T|P*pp) ₁Payoff can be determined in any of a number of different manners, suchas in a manner similar to that explained above with reference toperforming a multi-stage option valuation (see equations (6) and (6a)).The net conditional payoff may be calculated or otherwise determined byaccounting for the payout price at the payout point p₁ (e.g., t=4) andagain accounting for the exercise price at the expiration exercisepoint, now representing a price for exercising the option at the payoutpoint (see block 38 d). In addition, however, the net conditional payoffmay further account for the exercise prices of any exercise pointscoinciding with the payout point and/or between the payout point and theexpiration exercise point. Written notationally, for example, the netconditional payoff Net S_(T|P*pp) ₁ Payoff may be determined as follows:

$\begin{matrix}{{{Net}\mspace{11mu} S_{T|{P^{*}{pp}_{1}}}{Payoff}} = {{S_{T|{P^{*}{pp}_{1}}}{Payoff}} - \left( {{P^{*}{pp}_{1}{\mathbb{e}}^{{- r_{1}}{pp}_{1}}} - {x_{T}{\mathbb{e}}^{{- r_{2}}{pp}_{1}}}} \right) - {\sum\limits_{{ep}_{n^{\prime}} \geq {pp}_{1}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}} & (49)\end{matrix}$where x_(ep) _(n′) represents the exercise price (e.g., contingentfuture investment) at exercise point ep_(n′).

From the net conditional payoff, the mean net conditional payoff may bedetermined (see block 38 e). For example, the mean net conditionalpayoff Mean Net S_(T|P*pp) ₁ Payoff may be determined by selecting orotherwise forecasting a number of conditional asset values s_(T|P*pp) ₁from the conditional distribution of contingent future benefitsS_(T|P*pp) ₁ ; calculating, for those forecasted asset values,conditional payoff and net conditional payoff values such as inaccordance with equations (6a) and (49); and calculating or otherwisedetermining the mean of the calculated net conditional payoff values.The aforementioned steps thereby effectuating equations (6) and (49),including the expected value expression of equation (6). Then, afterdetermining the mean net conditional payoff value Mean Net S_(T|P*p) ₁Payoff, if the mean net conditional payoff value does not equalapproximately zero, another milestone threshold P*pp₁ may be estimatedfor the payout point P*pp₁ (see block 38 f). The method may then repeatdetermining a conditional distribution of contingent future benefitsS_(T|P*pp) ₁ , determining a conditional payoff S_(T|P*pp) ₁ Payoff, netconditional payoff Net S_(T|P*pp) ₁ Payoff and mean net conditionalpayoff Mean Net S_(T|P*pp) ₁ Payoff, and determining if the mean netconditional payoff value equals approximately zero. The method maycontinue in this manner until an estimated milestone threshold P*pp₁results in a mean net conditional payoff value equal to approximatelyzero. This estimated milestone threshold may then be considered themilestone threshold for the respective payout point.

2. Arc Technique for Determining Milestone Thresholds

Determining the milestone threshold for each exercise point P*ep_(n′)according to the benefit-regret technique of one exemplary embodiment ofthe present invention may proceed in a manner similar to that for themulti-stage option. Thus, and generally for each exercise point ep_(n′),n′=1, 2, . . . N′−1, a number of candidate milestone thresholds may beidentified P*ep_(n′,j) (see FIG. 2 b, block 18 h). Having selected thecandidate milestone thresholds P*ep_(n′,j), payoffs (or intermediatepayoffs) may then be determined (see block 18 i). In this regard, thepayoffs (or intermediate payoffs) may be determined as the expectedvalue of the difference between a present value distribution ofcontingent future benefits at the expiration exercise point (discountedby r₁), and the present value of the exercise price (discounted by r₂)at the expiration exercise point, and including limiting the minimumpermissible difference to a predefined value, such as zero. And for eachexercise point preceding the next-to-last exercise point ep_(N′−1),determining the payoffs may further include reducing the expected valuesof the difference (intermediate payoffs) by the present values of theexercise prices (discounted by r₂) at the one or more exercise pointsbetween the expiration exercise point and the respective exercise point.However, determining the payoffs may be further conditioned on the assetvalue at the respective exercise point and one or more, if not all,subsequent exercise points before the expiration exercise point S_(ep)_(n′) being at least as much as, if not greater than (i.e., ≧), thecandidate milestone threshold at the respective exercise pointsP*ep_(n′,j), such as by setting the payoffs to zero when the asset valueat an exercise point is less than the milestone threshold at therespective exercise point. Thus, written notationally, the payoffs maybe determined or otherwise calculated in accordance with the following:

$\begin{matrix}{{{{{IF}\mspace{14mu} S_{{ep}_{n^{\prime}}}} \geq {P^{*}{ep}_{n^{\prime},j}}},{\forall_{m^{\prime}}\left( {S_{p_{m^{\prime}}} \geq {P^{*}{ep}_{m^{\prime}}}} \right)},{then}}\mspace{20mu}{{\left. {S_{T}{Payoff}} \middle| {P^{*}{ep}_{n^{\prime},j}} \right. = {{E\left\lbrack {\max\left( {{{S_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack} - {\sum\limits_{{ep}_{m^{\prime}}}{x_{{ep}_{m^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{m^{\prime}}}}}}};}{{else},\text{}\mspace{20mu}{\left. {S_{T}{Payoff}} \middle| {P^{*}{ep}_{n^{\prime},j}} \right. = 0}}} & (50)\end{matrix}$where m′=n′+1, n′+2, . . . N′−1, and p_(m′)=p_(n′+1), p_(n′+2), . . .p_(N′−1) (noting that (n′+1)>(N′−1) results in the empty sets m′=Ø, andp_(m′)=Ø).And as a function of asset values s_(ep) _(n′) , s_(ep) _(m′) , ∀_(m′)and s_(T) from respective distributions S_(ep) _(n′) , S_(ep) _(m′) ,∀_(m′) and S_(T), the payoffs S_(T) Payoff|P*ep_(n′,j) may be determinedas follows:

$\begin{matrix}{{{{{IF}\mspace{14mu} s_{{ep}_{n^{\prime}}}} \geq {P^{*}{ep}_{n^{\prime},j}}},{\forall_{m^{\prime}}\left( {s_{{ep}_{m^{\prime}}} \geq {P^{*}{ep}_{m^{\prime}}}} \right)},{then}}\mspace{20mu}{{\left. {S_{T}{Payoff}} \middle| {P^{*}{ep}_{n^{\prime},j}} \right. = {{\max\left( {{{s_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} - {\sum\limits_{{ep}_{m^{\prime}}}{x_{{ep}_{m^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{m^{\prime}}}}}}};}{{else},\text{}\mspace{20mu}{\left. {S_{T}{Payoff}} \middle| {P^{*}{ep}_{n^{\prime},j}} \right. = 0}}} & \left( {50a} \right)\end{matrix}$

As or after determining the payoffs for an exercise point ep_(n),respective net payoffs at the respective exercise point may bedetermined or otherwise calculated, such as by reducing the payoffs bythe present value of the exercise price (discounted by r₂) at therespective exercise point (see block 18 j). Written notationally, thenet payoffs may be determined or otherwise calculated in accordance withthe following:Net S _(T) Payoff|P*ep _(n′,j) =S _(T) Payoff|P*ep _(n′,j) −x _(ep)_(n′) e ^(−r) ² ^(ep) ^(n′)   (51)From the net payoffs Net S_(T) Payoff|P*ep_(n′,j), respective mean netpayoffs Mean Net S_(T) Payoff|P*ep_(n′,j) may be determined forrespective candidate milestone thresholds P*ep_(n′,j) (see block 18 k).Again, for example, the mean net payoffs may be determined by firstselecting or otherwise forecasting a number of asset values s_(ep) _(n′), s_(ep) _(m′) , ∀_(m′) and s_(T) from respective distributions ofcontingent future value S_(ep) _(n′) , S_(ep) _(m′) , ∀_(m′) and S_(T).For those forecasted asset values, then, respective payoff and netpayoff values may be calculated such as in accordance with equations(50a) and (51). The means of the calculated net payoff values may thenbe calculated or otherwise determined, and a maximum mean net payoffvalue selected from the determined or otherwise calculated mean netpayoffs. The aforementioned steps being performed to thereby effectuateequations (50) and (51), including the expected value expression ofequation (51). The candidate milestone threshold associated with therespective mean net payoff value may then be selected as the milestonethreshold for the respective exercise point ep_(n).

Determining the milestone threshold for the payout point P*pp₁ accordingto the arc technique of one exemplary embodiment of the presentinvention may proceed in a manner similar to that for the early-launchoption. Thus, determining the milestone threshold may includeidentifying J candidate milestone thresholds for the payout pointP*pp_(1,j), j=1, 2, . . . J, and calculating or otherwise determiningrespective payoffs or profits based thereon (see FIG. 16 b, blocks 38 h,38 i). In this regard, the payoffs or profits may be conditioned onrespective candidate milestone thresholds, and determined in any of anumber of different manners, including in accordance with the DMalgorithm. In contrast to the early-launch case, however, the payoffsmay further account for the exercise prices of any exercise points.Written notationally, for example, the payoffs S_(T) Payoff|P*pp_(1,j)may be determined as follows:

$\begin{matrix}{{{{{IF}\mspace{14mu} S_{{pp}_{1}}} \geq {P^{*}{pp}_{1,j}}},{then}}\mspace{20mu}{{\left. {S_{T}{Payoff}} \middle| {P^{*}{pp}_{1,j}} \right. = {{S_{{pp}_{1}}{\mathbb{e}}^{{- r_{1}}{pp}_{1}}} - {x_{T}{\mathbb{e}}^{{- r_{2}}{pp}_{1}}} - {\sum\limits_{{ep}_{n^{\prime}} < {pp}_{1}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}};}{{else},\text{}\mspace{20mu}{\left. {S_{T}{Payoff}} \middle| {P^{*}{pp}_{1,j}} \right. = {{E\left\lbrack {\max\left( {{{S_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} \right\rbrack} - {\sum\limits_{{ep}_{n^{\prime}} < p_{N}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}}}} & (52)\end{matrix}$As above, x_(ep) _(n′) again represents the exercise price (e.g.,contingent future investment) at exercise point ep_(n′). And as afunction of asset values s_(pp) ₁ and s_(T) from respectivedistributions of contingent future value S_(pp) ₁ and S_(T), the payoffsS_(T) Payoff|P*pp_(1,j) may be determined as follows:

$\begin{matrix}{{{{{IF}\mspace{14mu} s_{{pp}_{1}}} \geq {P^{*}{pp}_{1,j}}},{then}}\mspace{20mu}{{\left. {S_{T}{Payoff}} \middle| {P^{*}{pp}_{1,j}} \right. = {{s_{{pp}_{1}}{\mathbb{e}}^{{- r_{1}}{pp}_{1}}} - {x_{T}{\mathbb{e}}^{{- r_{2}}{pp}_{1}}} - {\sum\limits_{{ep}_{n^{\prime}} < {pp}_{1}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}};}{{else},\text{}\mspace{20mu}{\left. {S_{T}{Payoff}} \middle| {P^{*}{pp}_{1,j}} \right. = {{\max\left( {{{s_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}},0} \right)} - {\sum\limits_{{ep}_{n^{\prime}} < p_{N}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}}}} & \left( {52a} \right)\end{matrix}$

As or after determining the payoffs at the expiration exercise point p₂(t=T), respective mean payoffs Mean S_(T) Payoff|P*pp_(1,j) may bedetermined for respective candidate milestone thresholds P*pp_(1,j) (seeblock 38 j). For example, the mean payoffs may be determined by firstselecting or otherwise forecasting a number of asset values s_(pp) ₁ ands_(T) from respective distributions of contingent future benefits S_(pp)₁ and S_(T). For those forecasted asset values, then, respective payoffvalues may be calculated such as in accordance with equation (52a). Themeans of the calculated payoff values may then be calculated orotherwise determined for respective candidate milestone thresholdsP*pp_(1,j). The aforementioned steps thereby effectuating equation (52),including its expected value expression. Thus, a maximum mean payoffvalue may then be selected from the determined or otherwise calculatedmean payoffs. The candidate milestone threshold associated with therespective mean payoff value may then be selected as the milestonethreshold for the payout point (see block 18 k).

Irrespective of exactly how the milestone thresholds P*p_(n) at theexercise points ep_(n′) and the payout point pp₁ before the expirationexercise point p_(N) are estimated or otherwise determined, the value ofthe combination option may thereafter be determined or otherwisecalculated based thereon. The value of the combination option may bedetermined in any of a number of different manners. Again, for example,the value of the combination option may be considered a payoff S_(T)Payoff, which may be determined based upon the previously determineddistributions of contingent future value S_(p) _(n) , n=1, 2, . . . N,as shown in block 60. And more particularly, the value of thecombination option may represent the payoff should the participantexercise the contingent claim at the payout point (after havingexercised contingent claims at exercise points up to the respectivepayout point), or the payoff should the participant forego earlyexercise of the option and instead exercise the contingent claim at theexpiration exercise point (again, after having exercised contingentclaims at exercise points up to the expiration exercise point). In thisregard, the milestone threshold P*pp₁ for the payout point may againrepresent the minimum asset value (future benefits value) at which areasonably prudent participant will exercise the contingent claim earlyat that point.

Written notationally, for example, the payoff S_(T) Payoff may bedetermined as follows:

$\begin{matrix}{{{{IF}\mspace{14mu}{\exists_{p_{n} = {pp}_{1}}\left( {S_{p_{n}} \geq {P^{*}{pp}_{1}}} \right)}},{then}}\mspace{76mu}{{{S_{T}{Payoff}} = {{S_{p_{n}}{\mathbb{e}}^{{- r_{1}}{pp}_{1}}} - {x_{T}{\mathbb{e}}^{{- r_{2}}{pp}_{1}}} - {\sum\limits_{{ep}_{n^{\prime}} < {pp}_{1}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}};}{{else},\text{}\mspace{65mu}{{IF}\mspace{14mu}{\forall_{p_{n} = {{ep}_{n^{\prime}} < p_{N}}}\left( {S_{p_{n}} \geq {P^{*}{ep}_{n^{\prime}}}} \right)}},{then}}\mspace{65mu}{{{S_{T}{Payoff}} = \mspace{95mu}{{E\left\lbrack {\max\left( {{{S_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}}, 0} \right)} \right\rbrack} - {\sum\limits_{{ep}_{n^{\prime}} < p_{N}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}};}\mspace{65mu}{{else},\text{}\mspace{59mu}{{S_{T}{Payoff}} = 0}}} & (53)\end{matrix}$And as a function of asset values s_(p) _(n) , ∀_(n<N) and s_(T) fromrespective distributions of contingent future value S_(p) _(n) , ∀_(n<N)and S_(T), the payoff S_(T) Payoff may be determined as follows:

$\begin{matrix}{{{{IF}\mspace{14mu}{\exists_{p_{n} = {pp}_{1}}\left( {s_{p_{n}} \geq {P^{*}{pp}_{1}}} \right)}},{then}}\mspace{65mu}{{{S_{T}{Payoff}} = {{s_{p_{n}}{\mathbb{e}}^{{- r_{1}}{pp}_{1}}} - {x_{T}{\mathbb{e}}^{{- r_{2}}{pp}_{1}}} - {\sum\limits_{{ep}_{n^{\prime}} < {pp}_{1}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}};}{{else},\text{}\mspace{59mu}{{IF}\mspace{14mu}{\forall_{p_{n} = {{ep}_{n^{\prime}} < p_{N}}}\left( {s_{p_{n}} \geq {P^{*}{ep}_{n^{\prime}}}} \right)}},{then}}\mspace{65mu}{{{S_{T}{Payoff}} = \mspace{56mu}{{\max\left( {{{s_{T}{\mathbb{e}}^{{- r_{1}}T}} - {x_{T}{\mathbb{e}}^{{- r_{2}}T}}}, 0} \right)} - {\sum\limits_{{ep}_{n^{\prime}} < p_{N}}{x_{{ep}_{n^{\prime}}}{\mathbb{e}}^{{- r_{2}}{ep}_{n^{\prime}}}}}}};}\mspace{59mu}{{else},\text{}\mspace{59mu}{{S_{T}{Payoff}} = 0}}} & \left( {53a} \right)\end{matrix}$

As or after determining the payoff for the combination option, the meanpayoff for the combination option (value of the combination option) maybe determined, such as by selecting or otherwise forecasting a number ofasset values s_(p) _(n) , ∀_(n<N) and s_(T) from respectivedistributions of contingent future value S_(p) _(n) , ∀_(n<N) and S_(T);calculating, for those forecasted asset values, payoff values such as inaccordance with equation (53a); and calculating or otherwise determiningthe mean of the calculated payoff values, such as in a manner similar tothat explained above, as shown in block 62. The aforementioned stepsthereby effectuating equation (53), including the expected valueexpression.

Written in other terms, the value of the combination option may beexpressed as follows:

$\begin{matrix}{E\left\lbrack {\ldots\mspace{11mu}{E_{S_{p_{N - 2}}}\lbrack}} \right.} \\{{\left. {{E_{S_{p_{N - 1}}}\left\lbrack {Z_{t_{0}}^{T}\left( {\mu,r} \right)} \right\rbrack} - {x_{p_{N - 1}}{\mathbb{e}}^{{- r_{2}}p_{N - 1}}}} \right\rbrack^{+} \times {p\left( {{P^{*}{pp}_{N - 1}} > s_{p_{N - 1}} > {P^{*}{ep}_{N - 1}}} \right)}} +} \\{{{E\left\lbrack {Z_{t_{0}}^{p_{N - 1}}\left( {\mu,r} \right)} \right\rbrack} \times {p\left( {s_{p_{N - 1}} > {P^{*}{pp}_{N - 1}}} \right)}} -} \\{{\left. {x_{p_{N - 2}}{\mathbb{e}}^{{- r_{2}}p_{N - 2}}} \right\rbrack^{+} \times {p\left( {{P^{*}{pp}_{N - 2}} > s_{p_{N - 2}} > {P^{*}{ep}_{N - 2}}} \right)}} +} \\{{{E\left\lbrack {Z_{t_{0}}^{p_{N - 2}}\left( {\mu,r} \right)} \right\rbrack} \times {p\left( {s_{p_{N - 2}} > {P^{*}{pp}_{N - 2}}} \right)}} -} \\\vdots \\{{{\left. {x_{p_{1}}{\mathbb{e}}^{{- r_{2}}p_{1}}} \right\rbrack^{+} \times {p\left( {{P^{*}{pp}_{1}} > s_{p_{1}} > {P^{*}{ep}_{1}}} \right)}} + {{E\left\lbrack {Z_{t_{0}}^{p_{1}}\left( {\mu,r} \right)} \right\rbrack} \times {p\left( {s_{p_{1}} > {P^{*}{pp}_{1}}} \right)}}};} \\{{n = {n^{\prime} = n^{''}}};{N = {N^{\prime} = N^{\prime\prime}}};} \\{{p_{n} = {{ep}_{n^{\prime}} = {pp}_{n^{\prime\prime}}}};} \\{{\exists{{P^{*}{pp}_{n^{\prime\prime}}\mspace{14mu}{iff}\mspace{14mu} y_{{pp}_{n^{''}}}} > 0}};} \\{\exists{{P^{*}{ep}_{n^{\prime}}\mspace{14mu}{iff}\mspace{14mu} x_{{ep}_{n^{\prime}}}} > 0}}\end{matrix}$The above expression may be interpreted as subtracting, from thediscounted payoff at the expiration exercise point (t=T), the discountedexercise prices for all of the exercise points before the expirationexercise point; and multiplying the resulting payoff by the probabilitythat, for the exercise points before the expiration exercise point, theasset values at the exercise points are greater than respectivemilestone thresholds for those exercise points. Also in the expressionthe “+” superscript represents a maximization function limiting thediscounted payoff to a minimum predefined value, such as zero. Inaddition, at each exercise point, there may be the option for anearly-launch, which may be interpreted as the sum of the value of theopportunity to launch early to preserve the total project or assetvalue, and the value of the waiting or delaying the launch decision(contingent claim) until the next exercise point. Each early-launchdecision payoff may be multiplied by the probability that, for theexercise points before the expiration exercise point, the asset valuesat the exercise points are greater than respective milestone thresholdsfor those exercise points. The cumulative probabilities for all decisionpoints sums to a value of one, including the probabilities of earlytermination of the project, i.e., the asset values at the exercisepoints are less than the respective milestone thresholds for thoseexercise points.

At each exercise point, one or both of a go, no-go decision or anearly-launch decision may be made available. Each additional decisiontype at each exercise point increases the flexibility of the entireproject proposition, and therefore increases the value of the totalcombination option. The type of decision available at each exercisepoint depends on the anticipated business environment and investmentplan. If there is no anticipation of an impaired project value at aparticular decision point, i.e., y_(n)=0, then P*pp_(n) for theearly-launch decision may approach infinity P*pp_(n)→∞, and may resultaccordingly be set or otherwise reduced to an empty set P*pp_(n)=Ø forthe respective decision point before the expiration exercise point.Additionally, if there is no investment required to proceed to the nextdecision point, i.e., x_(p) _(n) =0, then P*ep_(n) for the go, no-godecision may approach zero P*ep_(n)→0, and may result accordingly be setor otherwise reduced to an empty set P*ep_(n)=Ø for the respectivedecision point before the expiration exercise point.

From the above, the value of the combination option can be understood asthe value of a series of embedded expectations, with each expectationstage having the ability to either: (a) launch the project early in theevent of anticipated impairment of cash or asset value, or (b) continuethe project if the net payoff of the subsequent stage(s) or decisionpoint(s) is greater than zero, or (c) terminate the project if theassociated cost to continue to the subsequent stage is too costly. Theexpectations may arise because of the uncertainty of the payoff at eachstage, thus valued as a mean or expected value of the sum total of allpotential outcomes at each stage. The expectations may be embeddedbecause success (or net payoff greater than zero) at an earlier stageallows forward progress to the subsequent stage of the project. Failure(or net payoffs which are limited to zero) may terminate forwardprogress of the project. Milestone thresholds P*p_(n) (P*ep_(n) and/orP*pp_(n)) may be calculated for each stage or exercise point before theexpiration exercise point.

The value of the combination option may be the present value of theexpiration point payoffs less the cost of having traversed some or allthe intermediate decision points, and thus incurred a cost x_(p) _(n) tocontinue to a subsequent point weighted by the probability of actuallytraversing some or all of the intermediate decision points, plus thepresent value of an early-launch decision payoff weighted by theprobability or likelihood of the event of actually launching the projectearly at any one of the intermediate decision points.

As shown in FIG. 27, the system of the present invention is typicallyembodied by a processing element and an associated memory device, bothof which are commonly comprised by a computer 60 or the like. In thisregard, as indicated above, the method of embodiments of the presentinvention can be performed by the processing element manipulating datastored by the memory device with any one of a number of commerciallyavailable computer software programs. In one embodiment, the method canbe performed with data that is capable of being manipulated and/orpresented in spreadsheet form. For example, the method can be performedby the processing element manipulating data stored by the memory devicewith Excel, a spreadsheet software program distributed by the MicrosoftCorporation of Redmond, Wash., including Crystal Ball, a Monte Carlosimulation software program distributed by Decisioneering, Inc. ofDenver, Colo. The computer can include a display 62 for presentinginformation relative to performing embodiments of the method of thepresent invention, including the various distributions, models and/orconclusions as determined according to embodiments of the presentinvention. To plot information relative to performing embodiments of themethod of the present invention, the computer can further include aprinter 64.

Also, the computer 60 can include a means for locally or remotelytransferring the information relative to performing embodiments of themethod of the present invention. For example, the computer can include afacsimile machine 66 for transmitting information to other facsimilemachines, computers or the like. Additionally, or alternatively, thecomputer can include a modem 68 to transfer information to othercomputers or the like. Further, the computer can include an interface(not shown) to a network, such as a local area network (LAN), and/or awide area network (WAN). For example, the computer can include anEthernet Personal Computer Memory Card International Association(PCMCIA) card configured to transmit and receive information to and froma LAN, WAN or the like.

According to one aspect of the present invention, the system of thepresent invention generally operates under control of a computer programproduct according to another aspect of the present invention. Thecomputer program product for performing the methods of embodiments ofthe present invention includes a computer-readable storage medium, suchas the non-volatile storage medium, and computer-readable program codeportions, such as a series of computer instructions, embodied in thecomputer-readable storage medium.

In this regard, FIGS. 1, 2 a-2 d, 15, 16 a-16 d and 26 are flowcharts ofmethods, systems and program products according to exemplary embodimentsof the present invention. It will be understood that each block or stepof the flowchart, and combinations of blocks in the flowchart, can beimplemented by computer program instructions. These computer programinstructions may be loaded onto a computer or other programmableapparatus to produce a machine, such that the instructions which executeon the computer or other programmable apparatus create means forimplementing the functions specified in the flowchart block(s) orstep(s). These computer program instructions may also be stored in acomputer-readable memory that can direct a computer or otherprogrammable apparatus to function in a particular manner, such that theinstructions stored in the computer-readable memory produce an articleof manufacture including instruction means which implement the functionspecified in the flowchart block(s) or step(s). The computer programinstructions may also be loaded onto a computer or other programmableapparatus to cause a series of operational steps to be performed on thecomputer or other programmable apparatus to produce a computerimplemented process such that the instructions which execute on thecomputer or other programmable apparatus provide steps for implementingthe functions specified in the flowchart block(s) or step(s).

Accordingly, blocks or steps of the flowchart support combinations ofmeans for performing the specified functions, combinations of steps forperforming the specified functions and program instruction means forperforming the specified functions. It will also be understood that eachblock or step of the flowchart, and combinations of blocks or steps inthe flowchart, can be implemented by special purpose hardware-basedcomputer systems which perform the specified functions or steps, orcombinations of special purpose hardware and computer instructions.

Many modifications and other embodiments of the invention will come tomind to one skilled in the art to which this invention pertains havingthe benefit of the teachings presented in the foregoing descriptions andthe associated drawings. Therefore, it is to be understood that theinvention is not to be limited to the specific embodiments disclosed andthat modifications and other embodiments are intended to be includedwithin the scope of the appended claims. Although specific terms areemployed herein, they are used in a generic and descriptive sense onlyand not for purposes of limitation.

1. A system for performing a contingent claim valuation of an early-launch option including a contingent claim exercisable at one of a plurality of decision points that include an expiration exercise point and one or more decision points before the expiration exercise point, the system comprising: a processor configured to determine a first value representing a payoff attributable to exercise of a contingent claim at a selected decision point before the expiration exercise point, wherein the processor is configured to determine a second value representing a payoff attributable to the exercise of a contingent claim at the expiration exercise point, wherein the processor being configured to determine at least one of the first value or the second value includes being configured to determine at least one of the first value or the second value based upon a respective present value distribution of contingent future value and a respective present value of an exercise price, the present value distribution comprising a respective distribution of contingent future value discounted according to a first discount rate, and the present value of the exercise price comprising a respective exercise price discounted according to a second discount rate, and wherein the processor is configured to determine a value of the contingent claim based upon at least one of the first value or the second value.
 2. A system according to claim 1, wherein the processor is configured to determine the first value based upon a present value distribution of contingent future value at the selected decision point, and based upon a present value of an exercise price at the selected decision point.
 3. A system according to claim 1, wherein the processor is configured to determine the second value based upon a present value distribution of contingent future value at the expiration exercise point, and based upon a present value of an exercise price at the expiration exercise point.
 4. A system according to claim 3, wherein the processor being configured to determine a second value includes being configured to determine an expected value of the difference between the present value distribution of contingent future value at the expiration exercise point and the present value of the exercise price at the expiration exercise point.
 5. A system according to claim 4, wherein the processor being configured to determine the expected value of the difference includes being configured to limit the difference to a minimum predefined value.
 6. A system according to claim 1, wherein the processor is configured to determine the value further based upon a milestone threshold at the selected decision point, the milestone threshold representing a minimum asset value for exercising the contingent claim at the selected decision point.
 7. A system according to claim 6, wherein the processor being configured to determine a value includes being configured to condition the value on a comparison of a distribution of contingent future value at the selected decision point and the milestone threshold.
 8. A system according to claim 6, wherein the processor is further configured to determine the milestone threshold at the selected decision point, wherein the processor being configured to determine the milestone threshold includes being configured to: determine a present value conditional distribution of contingent future value attributable to the exercise of a contingent claim at the expiration exercise point, the present value conditional distribution comprising a distribution of contingent future value conditioned on an estimated milestone threshold, and discounted according to the first discount rate; determine present values of respective exercise prices required to exercise one or more contingent claims at the expiration exercise point and the selected decision point, the present values of respective exercise prices comprising respective exercise prices discounted according to the second discount rate; and determine a conditional value based upon the present value conditional distribution of contingent future value and the present values of respective exercise prices, wherein the processor is configured to repeatedly determine a present value conditional distribution and a conditional value for different estimated milestone thresholds until the conditional value equals approximately zero, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 9. A system according to claim 6, wherein the processor is further configured to determine the milestone threshold at the selected decision point, wherein the processor being configured to determine the milestone threshold includes being configured to: determine a first value and a second value; determine a value based upon at least one of the first value or the second value, the value being conditioned on a comparison of a distribution of contingent future value at the selected decision point and a candidate milestone threshold, wherein the processor is configured to repeatedly determine a value for a plurality of candidate milestone thresholds; and select a candidate milestone threshold that maximizes the value, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 10. A system according to claim 6, wherein the processor is further configured to determine the milestone threshold at the selected decision point, wherein the processor being configured to determine the milestone threshold includes being configured to: determine a present value of an underlying asset at a selected decision point before the expiration exercise point, the present value of the underlying asset comprising a candidate milestone threshold discounted according to the first discount rate; determine a third value based upon the present value of the underlying asset, and based upon a present value of an exercise price at the selected decision point, wherein the processing element processor is configured to determine a present value of an underlying asset and a first value for one or more candidate milestone thresholds for the selected decision point; and select a candidate milestone threshold for a third value approximately equal to a second value, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 11. A system according to claim 1, wherein the processor being configured to determine at least one of the first value or the second value based upon a respective present value of an exercise price includes being configured to determine at least one of the first value or the second value based upon a respective exercise price discounted according to a second discount rate that is unequal to the first discount rate.
 12. A method according to claim 1, wherein determining at least one of the first value or the second value based upon a respective present value of an exercise price comprises determining at least one of the first value or the second value based upon a respective exercise price discounted according to a second discount rate that is unequal to the first discount rate.
 13. A method of performing a contingent claim valuation of an early-launch option including a contingent claim exercisable at one of a plurality of decision points that include an expiration exercise point and one or more decision points before the expiration exercise point, the method comprising: determining a first value representing a payoff attributable to exercise of a contingent claim at a selected decision point before the expiration exercise point; determining a second value representing a payoff attributable to the exercise of a contingent claim at the expiration exercise point, wherein determining at least one of the first value or the second value comprises determining at least one of the first value or the second value based upon a respective present value distribution of contingent future value and a respective present value of an exercise price, the present value distribution comprising a respective distribution of contingent future value discounted according to a first discount rate, and the present value of the exercise price comprising a respective exercise price discounted according to a second discount rate; and determining a value of the contingent claim based upon at least one of the first value or the second value, wherein at least determining a first value and determining a second value are performed by a processor configured to perform at least determining a first value and determining a second value.
 14. A method according to claim 13, wherein determining a first value comprises determining a first value based upon a present value distribution of contingent future value at the selected decision point, and based upon a present value of an exercise price at the selected decision point.
 15. A method according to claim 13, wherein determining a second value comprises determining a second value based upon a present value distribution of contingent future value at the expiration exercise point, and based upon a present value of an exercise price at the expiration exercise point.
 16. A method according to claim 15, wherein determining a second value comprises determining an expected value of the difference between the present value distribution of contingent future value at the expiration exercise point and the present value of the exercise price at the expiration exercise point.
 17. A method according to claim 16, wherein determining an expected value of the difference includes limiting the difference to a minimum predefined value.
 18. A method according to claim 13, wherein determining a value comprises determining a value further based upon a milestone threshold at the selected decision point, the milestone threshold representing a minimum asset value for exercising the contingent claim at the selected decision point.
 19. A method according to claim 18, wherein determining a value includes conditioning the value on a comparison of a distribution of contingent future value at the selected decision point and the milestone threshold.
 20. A method according to claim 18 further comprising determining the milestone threshold at the selected decision point, wherein determining the milestone threshold comprises: determining a present value conditional distribution of contingent future value attributable to the exercise of a contingent claim at the expiration exercise point, the present value conditional distribution comprising a distribution of contingent future value conditioned on an estimated milestone threshold, and discounted according to the first discount rate; determining present values of respective exercise prices required to exercise one or more contingent claims at the expiration exercise point and the selected decision point, the present values of respective exercise prices comprising respective exercise prices discounted according to the second discount rate; and determining a conditional value based upon the present value conditional distribution of contingent future value and the present values of respective exercise prices, wherein determining a present value conditional distribution and a conditional value repeatedly occur for different estimated milestone thresholds until the conditional value equals approximately zero, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 21. A method according to claim 18 further comprising determining the milestone threshold at the selected decision point, wherein determining the milestone threshold comprises: determining a first value and a second value; determining a value based upon at least one of the first value or the second value, the value being conditioned on a comparison of a distribution of contingent future value at the selected decision point and a candidate milestone threshold, wherein determining a value repeatedly occurs for a plurality of candidate milestone thresholds; and selecting a candidate milestone threshold that maximizes the value, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 22. A method according to claim 18 further comprising determining the milestone threshold at the selected decision point, wherein determining the milestone threshold comprises: determining a present value of an underlying asset at a selected decision point before the expiration exercise point, the present value of the underlying asset comprising a candidate milestone threshold discounted according to the first discount rate; determining a third value based upon the present value of the underlying asset, and based upon a present value of an exercise price at the selected decision point, wherein determining a present value of an underlying asset and a first value occur for one or more candidate milestone thresholds for the selected decision point; and selecting a candidate milestone threshold for a third value approximately equal to a second value, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 23. A computer program product for performing a contingent claim valuation of an early-launch option including a contingent claim exercisable at one of a plurality of decision points that include an expiration exercise point and one or more decision points before the expiration exercise point, the computer program product comprising a computer-readable storage medium having computer-readable program code portions stored therein, the computer-readable program code portions comprising: a first executable portion configured to determine a first value representing a payoff attributable to exercise of a contingent claim at a selected decision point before the expiration exercise point; a second executable portion configured to determine a second value representing a payoff attributable to the exercise of a contingent claim at the expiration exercise point, wherein at least one of the first or second executable portions is configured to determine at least one of the first value or the second value based upon a respective present value distribution of contingent future value and a respective present value of an exercise price, the present value distribution comprising a respective distribution of contingent future value discounted according to a first discount rate, and the present value of the exercise price comprising a respective exercise price discounted according to a second discount rate; and a third executable portion configured to determine a value of the contingent claim based upon at least one of the first value or the second value.
 24. A computer program product according to claim 23, wherein the first executable portion is configured to determine the first value based upon a present value distribution of contingent future value at the selected decision point, and based upon a present value of an exercise price at the selected decision point.
 25. A computer program product according to claim 23, wherein the second executable portion is configured to determine the second value based upon a present value distribution of contingent future value at the expiration exercise point, and based upon a present value of an exercise price at the expiration exercise point.
 26. A computer program product according to claim 25, wherein the second executable portion being configured to determine a second value includes being configured to determine an expected value of the difference between the present value distribution of contingent future value at the expiration exercise point and the present value of the exercise price at the expiration exercise point.
 27. A computer program product according to claim 26, wherein the second executable portion being configured to determine the expected value of the difference includes being configured to limit the difference to a minimum predefined value.
 28. A computer program product according to claim 23, wherein the third executable portion is configured to determine the value further based upon a milestone threshold at the selected decision point, the milestone threshold representing a minimum asset value for exercising the contingent claim at the selected decision point.
 29. A computer program product according to claim 28, wherein the third executable portion being configured to determine a value includes being configured to condition the value on a comparison of a distribution of contingent future value at the selected decision point and the milestone threshold.
 30. A computer program product according to claim 28 further comprising: a fourth executable portion configured to determine the milestone threshold at the selected decision point, wherein the fourth executable portion being configured for determining the milestone threshold includes being configured to: determine a present value conditional distribution of contingent future value attributable to the exercise of a contingent claim at the expiration exercise point, the present value conditional distribution comprising a distribution of contingent future value conditioned on an estimated milestone threshold, and discounted according to the first discount rate; determine present values of respective exercise prices required to exercise one or more contingent claims at the expiration exercise point and the selected decision point, the present values of respective exercise prices comprising respective exercise prices discounted according to the second discount rate; and determine a conditional value based upon the present value conditional distribution of contingent future value and the present values of respective exercise prices, wherein the fourth executable portion is configured to repeatedly determine a present value conditional distribution and a conditional value for different estimated milestone thresholds until the conditional value equals approximately zero, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 31. A computer program product according to claim 28 further comprising: a fourth executable portion configured to determine the milestone threshold at the selected decision point, wherein the fourth executable portion being configured to determine the milestone threshold includes being configured to: determine a first value and a second value; determine a value based upon at least one of the first value or the second value, the value being conditioned on a comparison of a distribution of contingent future value at the selected decision point and a candidate milestone threshold, wherein the fourth executable portion is configured to repeatedly determine a value for a plurality of candidate milestone thresholds; and select a candidate milestone threshold that maximizes the value, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 32. A computer program product according to claim 28 further comprising: a fourth executable portion configured to determine the milestone threshold at the selected decision point, wherein the fourth executable portion being configured to determine the milestone threshold includes being configured to: determine a present value of an underlying asset at a selected decision point before the expiration exercise point, the present value of the underlying asset comprising a candidate milestone threshold discounted according to the first discount rate; determine a third value based upon the present value of the underlying asset, and based upon a present value of an exercise price at the selected decision point, wherein the fourth executable portion is configured to determine a present value of an underlying asset and a first value for one or more candidate milestone thresholds for the selected decision point; and select a candidate milestone threshold for a third value approximately equal to a second value, the respective milestone threshold being selected as the milestone threshold for the selected decision point.
 33. A computer program product according to claim 23, wherein at least one of the first or second executable portions being configured to determine at least one of the first value or the second value based upon a respective present value of an exercise price includes being configured to determine at least one of the first value or the second value based upon a respective exercise price discounted according to a second discount rate that is unequal to the first discount rate. 